n_{G}}\Bigr{)}\Bigr{]}\times\sum_{a=1}^{|\mathcal{S}|}{\mathcal{N}^{(a)}_{\mathfrak{n}}(-{1\over\tau},{z\over\tau})\widehat{\mathcal{N}}^{(a)}_{\mathfrak{n}}(-{1\over
c\tau},{z\over
c\tau})\over\mathcal{D}_{\mathfrak{n}}(-{1\over\tau},{z\over\tau})\widehat{\mathcal{D}}_{\mathfrak{n}}(-{1\over
c\tau},{z\over c\tau})}.$ (127)

As we explained in (98), the sum of $\mathfrak{i}$ and
${\hat{\mathfrak{i}}\over n_{G}}$ yields the $\Gamma_{0}(n_{G})$ index of the
elliptic genus. Therefore, the above expression can be written as follows,

$\displaystyle\mathbb{E}_{\mathfrak{n}}(\tau,z)$
$\displaystyle=\exp\Bigr{[}-{1\over 2\pi
i\tau}\Bigr{(}{\epsilon_{1}\epsilon_{2}\over 2}\mathfrak{n}^{T}(\Omega
D)\mathfrak{n}+(m^{2}-\epsilon_{+}^{2})\vec{D}\cdot\mathfrak{n}\Bigr{)}\Bigr{]}$
$\displaystyle\times\sum_{a=1}^{|\mathcal{S}|}{\mathcal{N}^{(a)}_{\mathfrak{n}}(-{1\over\tau},{z\over\tau})\widehat{\mathcal{N}}^{(a)}_{\mathfrak{n}}(-{1\over
c\tau},{z\over
c\tau})\over\mathcal{D}_{\mathfrak{n}}(-{1\over\tau},{z\over\tau})\widehat{\mathcal{D}}_{\mathfrak{n}}(-{1\over
c\tau},{z\over c\tau})}.$ (128)

Now, the remaining task is to determine the asymptotics of the second line of
(128) in the Cardy limit $\tau\to i\cdot 0^{+}$. In the expression, the
numerators and the denominators contain the following fugacities,

$\displaystyle Q_{1}^{D}=e^{-{\epsilon_{1}\over\tau}},\quad
Q_{2}^{D}=e^{-{\epsilon_{2}\over\tau}},\quad Q_{\tau}^{D}=e^{-{2\pi
i\over\tau}},\quad Q_{m}^{D}=e^{-{m\over\tau}},$
$\displaystyle\widehat{Q}_{1}^{D}=e^{-{\epsilon_{1}\over
n_{G}\tau}},\quad\widehat{Q}_{2}^{D}=e^{-{\epsilon_{2}\over
n_{G}\tau}},\quad\widehat{Q}_{\tau}^{D}=e^{-{2\pi i\over
n_{G}\tau}},\quad\widehat{Q}_{m}^{D}=e^{-{m\over n_{G}\tau}}.$ (129)

In our parameter setting (124), one can check that
$|Q_{1,2}^{D}|=|\hat{Q}_{1,2}^{D}|=1$ and
$|Q_{\tau}^{D}|,|\hat{Q}_{\tau}^{D}|\ll 1$. The other fugacities $Q_{m}^{D}$
and $\hat{Q}_{m}^{D}$ can also have extreme value depending on $\text{Im}[m]$.
For simplicity, let us assume that $m$ is purely imaginary and is in the
‘canonical chamber’ given as,

$\displaystyle 0<\text{Im}[m]<2\pi\quad\to\quad Q_{\tau}^{D}\ll Q_{m}^{D}\ll
1\ \&\ \widehat{Q}_{\tau}^{D}\ll\widehat{Q}_{m}^{D}\ll 1.$ (130)

Then, it is straightforward to obtain the asymptotics of (128). Our basis of
the elliptic genus is given in (94). Among those functions, we shall only
focus on the ones with $m$. After S-duality, their asymptotics are given as
follows,

$\displaystyle A(-{1\over\tau},{m\over\tau})$
$\displaystyle=(Q_{m}^{D})^{-1}\times\Bigr{(}1+\mathcal{O}(Q_{m}^{D},Q_{\tau}^{D}/Q_{m}^{D})\Bigr{)}$
$\displaystyle B(-{1\over\tau},{m\over\tau})$
$\displaystyle=(Q_{m}^{D})^{-1}\times\Bigr{(}1+\mathcal{O}(Q_{m}^{D},Q_{\tau}^{D}/Q_{m}^{D})\Bigr{)}$
$\displaystyle A(-{1\over n_{G}\tau},{m\over\tau})$
$\displaystyle=(\widehat{Q}_{m}^{D})^{-1}\times\Bigr{(}1+\mathcal{O}(\widehat{Q}_{m}^{D},\widehat{Q}_{\tau}^{D}/\widehat{Q}_{m}^{D})\Bigr{)}$
$\displaystyle B(-{1\over n_{G}\tau},{m\over\tau})$
$\displaystyle=(\widehat{Q}_{m}^{D})^{-1}\times\Bigr{(}1+\mathcal{O}(\widehat{Q}_{m}^{D},\widehat{Q}_{\tau}^{D}/\widehat{Q}_{m}^{D})\Bigr{)}.$
(131)

For the index $\mathfrak{i}_{\mathfrak{n}}$ of the $Z_{\mathfrak{n}}$, a
single $A(\tau,m)$ or $B(\tau,m)$ contributes $m^{2}$, and a single
$A(n_{G}\tau,m)$ or $B(n_{G}\tau,m)$ contributes ${m^{2}\over n_{G}}$. Since
the $m$-dependent part in $\mathfrak{i}_{\mathfrak{n}}$ is
$m^{2}D\cdot\mathfrak{n}$, one can conclude the following asymptotics,

$\displaystyle{\mathcal{N}^{(a)}_{\mathfrak{n}}(-{1\over\tau},{z\over\tau})\widehat{\mathcal{N}}^{(a)}_{\mathfrak{n}}(-{1\over
n_{G}\tau},{z\over
n_{G}\tau})\over\mathcal{D}_{\mathfrak{n}}(-{1\over\tau},{z\over\tau})\widehat{\mathcal{D}}_{\mathfrak{n}}(-{1\over
n_{G}\tau},{z\over
n_{G}\tau})}=(Q_{m}^{D})^{-D\cdot\mathfrak{n}}\times\Bigr{(}1+\mathcal{O}(\widehat{Q}_{m}^{D},\widehat{Q}_{\tau}^{D}/\widehat{Q}_{m}^{D})\Bigr{)}.$
(132)

Now, we plug the above result into (128). Then, one can obtain the following
Cardy limit asymptotics of the elliptic genus,

$\displaystyle\mathbb{E}_{\mathfrak{n}}(\tau,z)$
$\displaystyle=\exp\Bigr{[}-{1\over 2\pi
i\tau}\Bigr{(}{\epsilon_{1}\epsilon_{2}\over 2}\mathfrak{n}^{T}(\Omega
D)\mathfrak{n}+[m(m-2\pi
i)+\mathcal{O}(\epsilon^{2})]\vec{D}\cdot\mathfrak{n}\Bigr{)}+\mathcal{O}(\widehat{Q}_{m}^{D},\widehat{Q}_{\tau}^{D}/\widehat{Q}_{m}^{D})\Bigr{]}$
$\displaystyle\sim\exp\Bigr{[}-{1\over 2\pi
i\tau}\Bigr{(}{\epsilon_{1}\epsilon_{2}\over 2}\mathfrak{n}^{T}(\Omega
D)\mathfrak{n}+[m(m-2\pi i)]\vec{D}\cdot\mathfrak{n}\Bigr{)}\Bigr{]}.$ (133)

Next, let us consider the full 6d index. We shall focus on the non-Abelian
part of the index given as follows,

$\displaystyle
Z=\sum_{\mathfrak{n}}\mathbb{E}_{\mathfrak{n}}{\mathbf{Q}}^{\mathfrak{n}}.$
(134)

The electric fugacity $\mathbf{Q}=e^{-\mathbf{v}}$ probes the tensor branch
moduli space of 6d theory. We will focus on the origin of the tensor branch,
where conformal symmetry is restored. In the conformal phase, we should set
$\mathbf{v}=0$, i.e. ${\mathbf{Q}}=1$. By using (133), the 6d index can be
written as follows in the Cardy limit,

$\displaystyle Z\sim\sum_{\mathfrak{n}}\exp\Bigr{[}-{1\over 2\pi
i\tau}\Bigr{(}{\epsilon_{1}\epsilon_{2}\over 2}\mathfrak{n}^{T}(\Omega
D)\mathfrak{n}+[m(m-2\pi
i)]\sum_{a=1}^{r}D_{aa}\mathfrak{n}_{a}\Bigr{)}\Bigr{]}.$ (135)

The summand has a Gaussian structure with respect to the string charge
$\mathfrak{n}$. Furthermore, the sign of the quadratic term in the exponent is
negative, which guarantees the convergence of the summation.

The summation over $\mathfrak{n}$ can be evaluated using the continuum
approximation. Let us consider a new variable below,

$\displaystyle\mathfrak{x}\equiv-\epsilon_{1}\epsilon_{2}D\mathfrak{n},\quad(x_{a}=-\epsilon_{1}\epsilon_{2}D_{aa}n_{a}).$
(136)

Note that the minimal discrete distance $\Delta\mathfrak{n}=1$ while
$\Delta\mathfrak{x}=-\epsilon_{1}\epsilon_{2}\ll 1$, and $\mathfrak{x}$ can be
though as an almost continuous variable. Then, the summation (135) can be
approximated as the following integral,

$\displaystyle
Z\sim\int_{0}^{\infty}d^{r}\mathfrak{x}\exp\Bigr{[}{1\over(-2\pi
i\tau)\epsilon_{1}\epsilon_{2}}\Bigr{(}{1\over
2}\mathfrak{x}^{T}(D^{-1}\Omega)\mathfrak{x}-m(m-2\pi i)(\mathfrak{x}^{T}\cdot
I)\Bigr{)}\Bigr{]}.$ (137)

Here, we ignored the Jacobian factor which is subleading than
${1\over\tau\epsilon_{1}\epsilon_{2}}$ terms in the exponent. Also, we
introduced $r$-dimensional identity vector $I=(1,1,...,1)$. The free energy
$\log Z$ can be computed by using the saddle point approximation of
$\mathfrak{x}$. The saddle point equation and its solution are given by

$\displaystyle(D^{-1}\Omega)\mathfrak{x}-m(m-2\pi
i)I=0\quad\to\quad\mathfrak{x}=m(m-2\pi i)(\Omega^{-1}D)I.$ (138)

By plugging the saddle point value (138) to (137), one can obtain the
following 6d free energy,

$\displaystyle\log Z\simeq-{I^{T}\cdot\Omega^{-1}D\cdot I\over 2}{m^{2}(2\pi
i-m)^{2}\over(-2\pi i\tau)\epsilon_{1}\epsilon_{2}},$ (139)

which is valid in the Cardy limit. For all Lie algebras, one can check that
the overall coefficient is equivalent to the following group theoretic
constant,

$\displaystyle I^{T}\cdot\Omega^{-1}D\cdot I={h_{G}d_{G}\over 12},$ (140)

where $d_{G}$ is the dimension and $h_{G}$ is the dual Coxeter number of $G$.
The values are listed in table 7. As a conclusion, the Cardy free energy of 6d
$(2,0)$ theory of type $G$ is given as follows,

$\displaystyle\log Z\sim-{h_{G}d_{G}\over 24}{m^{2}(2\pi i-m)^{2}\over(-2\pi
i\tau)\epsilon_{1}\epsilon_{2}},$ (141)

where the 6d theory is twisted if $G$ is non-simply-laced. Interestingly, one
can observe that the 6d Cardy free energy (141) (without Abelian contribution)
takes a universal form for all Lie algebras, even with the outer-automorphism
twist. For the classical algebras $A_{r}$, $B_{r}$, $C_{r}$, and $D_{r}$, the
free energy scales as $\log Z\sim r^{3}$ when $r\gg 1$. This reiterates the
fact that $N$ M5-branes have $N^{3}$ degrees of freedom in the large $N$ limit
Klebanov:1996un .

Furthermore, we can compare the Cardy free energy for theories before and
after twisting. From the table 7, one can see that for all the Langlands dual
pairs of Lie algebras $(G^{(n)},G^{\vee(1)})$ of the twisted compactification
in the table 1, the following identity holds:

$\frac{h_{G}\,d_{G}}{n_{G}}=h_{G^{\vee}}\,d_{G^{\vee}}.$ (142)

Plugging it into (141), we find that the Cardy free energy remains unchanged
after the twist, provided that we take the change of radius into account. This
nicely shows that in the large radius limit, the effect of twisting is not
visible and the physics should remain the same.

| $d_{G}$ | $h_{G}$ | $|\vec{\rho}|^{2}$
---|---|---|---
$A_{r}$ | $(r+1)^{2}-1$ | $r+1$ | ${1\over 12}r^{3}+{1\over 4}r^{2}+{1\over 6}r$
$B_{r}$ | $2r^{2}+r$ | $2r-1$ | ${1\over 3}r^{3}+{1\over 4}r^{2}-{1\over 12}r$
$C_{r}$ | $2r^{2}+r$ | $r+1$ | ${1\over 3}r^{3}+{1\over 4}r^{2}-{1\over 12}r$
$D_{r}$ | $2r^{2}-r$ | $2r-2$ | ${1\over 3}r^{3}-{1\over 2}r^{2}+{1\over 6}r$
$G_{2}$ | 14 | 4 | 8
$F_{4}$ | 52 | 9 | 55
$E_{6}$ | 78 | 12 | 78
$E_{7}$ | 133 | 18 | ${399\over 2}$
$E_{8}$ | 248 | 30 | 620
Table 7: dimension, dual Coxeter number, square of Weyl vector of Lie algebras

In the computation above, the non-zero value of the saddle point (138) has a
special importance. Physically, it indicates that the string charge
$\mathfrak{n}$ condensates to a non-zero expectation value in the Cardy limit
Lee:2020rns . From the saddle point of $\mathfrak{x}$ in (138), the string
charge $\mathfrak{n}$ in (136) can be written as follows,

$\displaystyle\langle n_{a}\rangle={m(2\pi
i-m)\over\epsilon_{1}\epsilon_{2}}\sum_{b=1}^{r}(D^{-1}\Omega^{-1}D)_{ab}\,,$
(143)

where $\langle n_{a}\rangle$ denotes the value of $n_{a}$ at the saddle point.
The interesting fact is that the above expression is identical to the Weyl
vector of the Lie algebra. To be specific,

$\displaystyle\sum_{b=1}^{r}(D^{-1}\Omega^{-1}D)_{ab}\vec{\alpha}_{b}={1\over
2}\sum_{\vec{\alpha}\in\Delta^{+}(G)}\vec{\alpha}=\vec{\rho}\,,$ (144)

where $\vec{\alpha}_{b}$ is the root vector of the $b$’th node of the Dynkin
diagram of $G$, and $\Delta^{+}(G)$ is the positive root system of $G$. The
half of the sum over the positive root vectors $\rho$ is called the Weyl
vector of $G$. As a result, for all 6d (2,0) theory with or without twist, the
string charge $\mathfrak{n}$ has the following value at the saddle point,

$\displaystyle\langle\mathfrak{n}\rangle={m(2\pi
i-m)\over\epsilon_{1}\epsilon_{2}}\vec{\rho}\,.$ (145)

As well as the free energy, the sum of the string charge
$\langle\sum_{a}n_{a}\rangle\sim|\vec{\rho}|^{2}$ also scales as $r^{3}$ when
$r\gg 1$. For the simply-laced cases, the Weyl vector sum precisely explains
the free energy, i.e. $|\vec{\rho}|^{2}={h_{G}^{\vee}d_{G}\over 12}$, while
they are different for the non-simply-laced cases. The values of
$|\vec{\rho}|^{2}$ are listed in table 7.

Lastly, let us make a brief comment on the periodicity of $m$. The Cardy
formula (141) is obtained in the canonical chamber (130) where
$0<\text{Im}[m]<2\pi$. One can do a similar computation outside of the
canonical chamber, but it needs modification on the asymptotic behavior of
(132). For $ADE$ -type theories, such computation was performed in Lee:2020rns
; Duan:2020cta , and the result restores the periodicity $m\sim m+2\pi i$. We
expect that the Cardy formulas for twisted $(2,0)$ theories are also periodic
under $m\sim m+2\pi i$.

##  6 Conclusions

In this paper, we studied 6d $(2,0)$ SCFTs on a circle with outer-automorphism
twist and their BPS partition functions. The twisted $(2,0)$ theories are the
UV fixed points of 5d $\mathcal{N}=2$ SYM with non-simply-laced gauge groups:
$B_{r}$, $C_{r}$, $G_{2}$, and $F_{4}$. We used blowup equations to obtain the
BPS spectra and could determine the elliptic genera of twisted M-strings with
modular bootstrap. Our results for the elliptic genera of twisted M-strings
are completely new since there has been no known method to compute them.
Lastly, we obtain universal 6d Cardy formulas for the twisted $(2,0)$
theories, which yields $\log Z\sim(\text{rank})^{3}$ when the tensor branch
rank is large.

In this paper, we also determine the modular property of the elliptic genera
of twisted M-strings. If we impose outer-automorphism twist $\mathbb{Z}_{n}$
on M-strings, its elliptic genus is not a modular form under
$\text{SL}(2,\mathbb{Z})$, but its congruence subgroup. The modular index of
the twisted M-string can be obtained by folding the original untwisted
M-string, as described in (42). This modular index was double-checked by
comparing with the elliptic blowup equation (86). As a result, we can
bootstrap the twisted elliptic genus from the ring structure of
$\Gamma_{0}(N)$.

There are several interesting topics that worth pursuing in the near future.
The first and most obvious one is to understand the 2d quiver theory living on
the twisted BPS string. In particular, for the (2,0) SCFTs of $A$ type one is
supposed to perform certain twist or projection on the known 2d quiver
Haghighat:2013gba . However, at present it is not clear to the authors what
are the correct rules, as the elliptic genera on two sides seem not related in
a simple way.

Also, one may consider the generalization of our results to 6d little string
theories (LSTs) Aharony:1999ks . The 6d LSTs are non-local quantum field
theories and enjoy T-duality. Specifically, $(2,0)$ LSTs are T-dual to $(1,1)$
LSTs Kim:2015gha ; Kim:2018gak . $(2,0)$ LSTs have well-known $ADE$
classification, but one can impose an outer-automorphism twist on a T-dual
circle. We speculate that the corresponding T-dual theory is a $(1,1)$ LST
with a non-simply-laced gauge group. It would be interesting if one can check
the duality by explicitly computing the elliptic genera of 6d twisted little
strings. Similarly, we can start with (1,1) LST with ADE group and compactify
with an automorphism twist on T-dual circle. The resulting (2,0) theories seem
to lie beyond the ADE classification. It would be interesting whether these
theories make any sense.

Another interesting direction is to explore the blowup equation for the little
string theories, in particular, the blowup equation for 6d $(1,1)$ LSTs. The
main difference between LSTs and SCFTs is that there is an additional non-
shrinkable circle, which may lead to non-local objects in the blowup
equations. However, notice that a 6d $(1,1)$ little string theory can be
regarded as an elliptic version of the 5d $\mathcal{N}=1^{*}$ theories, which
means the blowup equation for $(1,1)$ little strings is simply an elliptic
lift of (70). We hope to present it in the near future.

Twisted strings in 6d $(1,0)$ SCFTs are also interesting objects to study.
Twisted compactification of 6d SCFTs yields 5d KK theories, which are parents
of large class of 5d SCFTs Bhardwaj:2019fzv . It has been recently found that
the blowup equations are strong methods to obtain the BPS spectra of those 5d
theories on $\mathbb{R}^{4}\times S^{1}$ Kim:2020hhh . However, non-critical
strings in twisted 6d SCFTs have not been studied extensively so far. We
expect that the modular property of twisted strings in $(1,0)$ SCFTs can be
also obtained from folding procedure similar to (42), and the modular
bootstrap would be a useful tool to explore the physics on those strings.

###### Acknowledgements.

We are grateful to Amir-Kian Kashani-Poor, Hee-Cheol Kim, Sung-Soo Kim,
Albrecht Klemm and Kaiwen Sun for useful discussions. We would like to thank
Amir-Kian Kashani-Poor for his reading of the draft and providing inspiring
comments. ZD, JN, KL and XW are supported by KIAS Individual Grant PG076901,
PG076401, PG006904 and QP079201 respectively. KL is also supported in part by
the National Research Foundation of Korea Grant NRF-2017R1D1A1B06034369.

##  Appendix A Modular forms

In this appendix, we briefly summarize necessary results about the theory of
modular forms Zagierbook ; MR0344216 ; EZ .

### Elliptic Modular Forms

######  Definition 1.

A function $f:\mathbb{H}\rightarrow\mathbb{C}$ is called a modular form of
integer weight $k$ for a modular group $\Gamma$ if f is holomorphic on
$\mathbb{H}\cup\\{\infty\\}$ and satisfies the following equation

$f\left(\frac{a\tau+b}{c\tau+d}\right)=(c\tau+d)^{k}f(\tau),\ \mathrm{for\
any}\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)\in\Gamma$ (146)

For example, let us consider the full modular group $\text{SL}(2,\mathbb{Z})$.
We can choose the matrix $T=\begin{pmatrix}1&1\\\ 0&1\\\ \end{pmatrix}$ to
find that $f$ is periodic $f(\tau+1)=f(\tau)$. It is standard to introduce
$q=\exp(2\pi i\tau)$, such that $f$ can be expanded as a power series in $q$,

$f(\tau)=\sum\limits_{n\geq 0}a_{n}q^{n}\,.$ (147)

$f$ being holomorphic yields $n\geq 0$. If furthermore $a_{0}=0$, $f$ is
called a $\mathit{cusp}$ form.

Modular forms naturally form a ring $M_{\ast}(\Gamma)$, graded by the weight.
The important result is that such space is always finitely generated. For the
full modular group $\text{SL}(2,\mathbb{Z})$,
$M_{\ast}(\text{SL}(2,\mathbb{Z}))$ is freely generated by $E_{4}$ and $E_{6}$
Zagierbook ; MR0344216 , where $E_{k}$ is the famous Eisenstein series,

$E_{k}(\tau)=\frac{1}{\zeta(k)}\sum_{\begin{subarray}{c}(m,n)\in\mathbb{Z}^{2}\\\
(m,n)\neq(0,0)\end{subarray}}\frac{1}{(m+n_{G}\tau)^{k}}\,,\quad
E_{k}|_{q=0}=1-\frac{2k}{B_{k}}\sum\limits_{n=1}^{\infty}\sigma_{k-1}(n)\,q^{n}\,,$
(148)

with $B_{k}$ the $k^{\text{th}}$ Bernoulli number and $\sigma_{i}(n)$ the sum
of the $i^{\text{th}}$ powers of the positive divisors of $n$.

Moreover, in this paper it is equally important to consider congruence
subgroups $\Gamma_{0}(N)$ of $\text{SL}(2,\mathbb{Z})$, defined as follows,

$\displaystyle\Gamma_{0}(N)=\Bigr{\\{}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\text{SL}(2,\mathbb{Z}):c\equiv 0\ (\text{mod}\
N)\Bigr{\\}}\,.$ (149)

As a special case, $\Gamma_{0}(1)=\text{SL}(2,\mathbb{Z})$. The ring of
modular forms for $\Gamma_{0}(N)$ will be denoted simply as $M_{\ast}(N)$.
Since the group is smaller, it is possible to have more generators. We
introduce

$E_{2}^{(N)}(\tau)=-\frac{24}{N-1}\frac{1}{2\pi
i}\partial_{\tau}\log\left(\frac{\eta(\tau)}{\eta(n_{G}\tau)}\right)\,,$ (150)

where $\eta(\tau)$ is the Dedekind eta function
$\eta(\tau)=q^{\frac{1}{24}}\prod\limits_{i=1}^{\infty}(1-q^{i})$. It is not
difficult to show that $E_{2}^{(N)}$ is an element of weight two in
$M_{\ast}(N)$. In the main text, we need to consider $N=2,3,4$, for which we
have the following result,

$\displaystyle M_{\ast}(2)$ $\displaystyle=\langle
E_{2}^{(2)}(\tau),E_{4}(\tau)\rangle\,,$ (151) $\displaystyle M_{\ast}(3)$
$\displaystyle=\langle E_{2}^{(3)}(\tau),E_{4}(\tau),E_{6}(\tau)\rangle\,,$
$\displaystyle M_{\ast}(4)$ $\displaystyle=\langle
E_{2}^{(2)}(\tau),E_{2}^{(4)}(\tau),E_{4}(\tau),E_{6}(\tau)\rangle\,.$

However, in actual computations, we find it enough in mostly of the cases to
use the modular forms $\widehat{E}_{k}(\tau):=E_{k}(N\tau)$. It is not
difficult to show that $\widehat{E}_{k}(\tau)$ indeed belongs to
$M_{\ast}(N)$.

### Jacobi Modular Forms

######  Definition 2.

A Jacobi modular form of weight $k$ and index $m$ for a modular group $\Gamma$
is a function $\phi:\mathbb{H}\times\mathbb{C}\rightarrow\mathbb{C}$ that
depends on a modular parameter $\tau\in\mathbb{H}$ and an elliptic parameter
$z\in\mathbb{C}$. It transforms under the action of $\Gamma$ on
$\mathbb{H}\times\mathbb{C}$ as

$\tau\mapsto\tau_{\gamma}=\frac{a\tau+b}{c\tau+d},\quad z\mapsto
z_{\gamma}=\frac{z}{c\tau+d}\quad{\rm
with}\quad\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)\in\Gamma\ ,$
(152)

as

$\displaystyle\phi_{k,m}\left(\tau_{\gamma},z_{\gamma}\right)$
$\displaystyle=(c\tau+d)^{k}e^{\frac{2\pi
imcz^{2}}{c\tau+d}}\phi_{k,m}(\tau,z)\,,$ (153)
$\displaystyle\phi_{k,m}(\tau,z+\lambda\tau+\mu)$ $\displaystyle=e^{-2\pi
im(\lambda^{2}\tau+2\lambda
z)}\phi_{k,m}(\tau,z)\quad\forall\lambda,\mu\in\mathbb{Z}\,,$ (154)

Let us first consider $\Gamma$ to be $\text{SL}(2,\mathbb{Z})$. Choosing
$T=\begin{pmatrix}1&1\\\ 0&1\\\ \end{pmatrix}$ and $\lambda=0,\mu=1$ in
equations (153) and (154) respectively we learn that the Jacobi form is
invariant under both shifts $\tau\rightarrow\tau+1$ and $z\rightarrow z+1$, so
it’s legitimate to write down the expansion,

$\phi(\tau,z)_{k,m}=\sum_{n,r}c(n,r)q^{n}y^{r}\,,\quad{\rm with}\ \ q=e^{2\pi
i\tau},\ \ y=e^{2\pi iz}\,.$ (155)

The coefficient $c(n,r)$ actually only depends on $r$ and an
$\text{SL}(2,\mathbb{Z})$ invariant combination $4nm-r^{2}$. We are interested
in the so-called weak Jacobi modular forms, which satisfy $c(n,r)=0$ unless
$n\geq 0$. We denote its ring as $J^{w}_{\ast,\ast}(\Gamma)$.

It is shown in EZ  that the ring of weak Jacobi forms
$J^{w}_{\ast,\ast}(\text{SL}(2,\mathbb{Z}))$ with even weight and integral
index is freely generated over $M_{\ast}(\text{SL}(2,\mathbb{Z}))$ by two
generators $\varphi_{-2,1}(\tau,z)$ and $\varphi_{0,1}(\tau,z)$. They can be
defined in terms of Jacobi theta functions,

$\displaystyle A(\tau,z)$
$\displaystyle\equiv\varphi_{-2,1}(\tau,z)=\frac{\theta_{1}(\tau,z)^{2}}{\eta(\tau)^{6}}\,,$
(156) $\displaystyle B(\tau,z)$
$\displaystyle\equiv\varphi_{0,1}(\tau,z)=4\Big{(}\frac{\theta_{2}(\tau,z)^{2}}{\theta_{2}(\tau,0)^{2}}+\frac{\theta_{3}(\tau,z)^{2}}{\theta_{3}(\tau,0)^{2}}+\frac{\theta_{4}(\tau,z)^{2}}{\theta_{4}(\tau,0)^{2}}\Big{)}.$

By the same token, we also need to consider the Jacobi modular forms for the
congruence subgroups $J^{w}_{\ast,\ast}(\Gamma_{0}(N))$. For us, the most
important class of examples is the Jacobi forms $\phi_{k,m}(N\tau,z)$ where
$\phi_{k,m}(\tau,z)$ belongs to $J^{w}_{\ast,\ast}(\text{SL}(2,\mathbb{Z}))$.
For brevity we just denote it by $\widehat{\phi}_{k,m}$, with $N$ understood
from the context. Also note that the actual index of $\phi_{k,m}(\tau,z)$ is
reduced to $m/N$, as is shown explicitly in the section 4.1.

In the final part, let us show the identities among the generators that are
used in our modular ansatz, for $N=2,3$. For modular forms involving $\tau$
and $2\tau$, we find the following relations,

$\displaystyle\widehat{E}_{4}$
$\displaystyle=\frac{5}{4}\left(E^{(2)}_{2}\right)^{2}-\frac{1}{4}E_{4}\,,$
(157) $\displaystyle\widehat{E}_{6}$
$\displaystyle=\frac{7}{8}\left(E^{(2)}_{2}\right)^{3}+\frac{1}{8}E_{6}\,,$
$\displaystyle A$
$\displaystyle=\frac{1}{12}\widehat{A}\widehat{B}+\frac{1}{12}E^{(2)}_{2}\widehat{A}^{2}\,,$
$\displaystyle B$
$\displaystyle=\frac{1}{12}\widehat{B}\left(E^{(2)}_{2}\widehat{A}+\widehat{B}\right)-\frac{1}{16}\widehat{A}^{2}\left(\left(E^{(2)}_{2}\right)^{2}-E_{4}\right)\,.$

In order to prove them, we need to invoke the following known identities
involving the Jacobi theta functions,

$\displaystyle\eta(\tau)^{3}$
$\displaystyle=\frac{1}{2}(\theta_{2}\theta_{3}\theta_{4})\,,$ (158)
$\displaystyle\theta_{3}^{4}$ $\displaystyle=\theta_{2}^{4}+\theta_{4}^{4}\,,$
$\displaystyle E^{(2)}_{2}$
$\displaystyle=\frac{1}{2}\theta_{3}^{4}+\frac{1}{2}\theta_{4}^{4}\,,$
$\displaystyle E_{4}$
$\displaystyle=\frac{1}{2}(\theta_{2}^{8}+\theta_{3}^{8}+\theta_{4}^{8})\,,$
$\displaystyle E_{6}$
$\displaystyle=\frac{1}{2}(-3\theta_{2}^{8}(\theta_{3}^{4}+\theta_{4}^{4})+\theta_{3}^{12}+\theta_{4}^{12})\,,$
$\displaystyle\widehat{\theta}_{2}^{2}$
$\displaystyle=\frac{1}{2}\theta_{3}^{2}-\frac{1}{2}\theta_{4}^{2}\,,$
$\displaystyle\widehat{\theta}_{3}^{2}$
$\displaystyle=\frac{1}{2}\theta_{3}^{2}+\frac{1}{2}\theta_{4}^{2}\,,$
$\displaystyle\widehat{\theta}_{4}^{2}$
$\displaystyle=\theta_{3}\theta_{4}\,,$
$\displaystyle\widehat{\theta}_{1}(z)\widehat{\theta}_{4}(z)$
$\displaystyle=\frac{1}{2}\theta_{2}\theta_{1}(z)\,,$
$\displaystyle\widehat{\theta}_{1}(z)^{2}+\widehat{\theta}_{4}(z)^{2}$
$\displaystyle=\theta_{3}\theta_{4}(z)\,.$

As an example, we present here a proof of the last equality in (157).

RHS
$\displaystyle=\frac{A}{\widehat{A}}\widehat{B}+\frac{3}{64}\theta_{2}^{8}\widehat{A}^{2}\,,$
(159)
$\displaystyle=\frac{A}{\widehat{A}}\left[\widehat{A}\left(3\widehat{\theta}_{3}^{2}\widehat{\theta}_{2}^{2}\frac{\widehat{\theta}_{4}^{2}(z)}{\widehat{\theta}_{1}^{2}(z)}-\widehat{\theta}_{3}^{4}-\widehat{\theta}_{2}^{4}\right)\right]+\frac{3}{64}\theta_{2}^{8}\frac{\widehat{\theta}_{1}^{4}(z)}{\widehat{\eta}^{12}}\,,$
$\displaystyle=A\left[\frac{3}{4}\theta_{2}^{4}\left(\frac{\theta_{3}\theta_{4}(z)}{\widehat{\theta}_{1}^{2}(z)}-1\right)-\theta_{3}^{4}+\frac{1}{2}\theta_{2}^{4}\right]+\frac{3}{64}\theta_{2}^{8}\frac{\widehat{\theta}_{1}^{4}(z)}{\widehat{\eta}^{12}}\,,$
$\displaystyle=A\left[\frac{3}{4}\theta_{2}^{4}\left(\frac{4\theta_{3}^{2}\theta_{4}^{2}(z)-4\theta_{3}\theta_{4}(z)\widehat{\theta}_{1}^{2}(z)}{\theta_{1}^{2}(z)\theta_{2}^{2}}-1\right)-\theta_{3}^{4}+\frac{1}{2}\theta_{2}^{4}\right]+12\frac{\widehat{\theta}_{1}^{4}(z)}{\theta_{3}^{2}\theta_{4}^{2}}\,,$
$\displaystyle=A\left(3\frac{\theta_{3}^{2}\theta_{3}^{2}\theta_{4}^{2}(z)}{\theta_{1}^{2}(z)}-\theta_{3}^{4}-\frac{1}{4}\theta_{2}^{4}\right)-12\frac{\theta_{4}(z)\widehat{\theta}_{1}^{2}(z)}{\theta_{3}\theta_{4}^{2}}+12\frac{\widehat{\theta}_{1}^{2}(z)(\theta_{3}\theta_{4}(z)-\widehat{\theta}_{4}^{2}(z))}{\theta_{3}^{2}\theta_{4}^{2}}\,,$
$\displaystyle=A\left(3\frac{\theta_{3}^{2}\theta_{3}^{2}\theta_{4}^{2}(z)}{\theta_{1}^{2}(z)}-\theta_{3}^{4}-\frac{1}{4}\theta_{2}^{4}\right)-3\frac{\theta_{2}^{2}\theta_{1}^{2}(z)}{\theta_{3}^{2}\theta_{4}^{2}}\,,$
$\displaystyle=A\left(3\frac{\theta_{3}^{2}\theta_{3}^{2}\theta_{4}^{2}(z)}{\theta_{1}^{2}(z)}-\theta_{3}^{4}-\frac{1}{4}\theta_{2}^{4}\right)-\frac{3}{4}A\theta_{2}^{4}\,,$
$\displaystyle=\text{LHS}\,.$

For modular forms involving $\tau$ and $3\tau$, the situation becomes more
complicated. Experimentally, we find the following identities which we verify
up to very high orders in $q$,

$\displaystyle\widehat{E}_{4}$
$\displaystyle=\frac{1}{9}\left(10\left(E^{(3)}_{2}\right)^{2}-E_{4}\right)\,,$
(160) $\displaystyle\widehat{E}_{6}$
$\displaystyle=\frac{1}{27}\left(35\left(E^{(3)}_{2}\right)^{3}-7E_{4}E^{(3)}_{2}-E_{6}\right)\,,$
$\displaystyle A$
$\displaystyle=\frac{1}{144}\widehat{A}\left(\left(E^{(3)}_{2}\right)^{2}\widehat{A}^{2}+2E^{(3)}_{2}\widehat{A}\widehat{B}+\widehat{B}^{2}\right)\,,$
$\displaystyle B$
$\displaystyle=-\frac{1}{1944}\widehat{A}^{3}{\left(23\left(E^{(3)}_{2}\right)^{3}-19E_{4}E^{(3)}_{2}-4E_{6}\right)}\,$
$\displaystyle\ \ \
+\frac{1}{432}\widehat{B}\left[\left(E^{(3)}_{2}\right)^{2}\widehat{A}^{2}+2E_{4}\widehat{A}^{2}+6E^{(3)}_{2}\widehat{A}\widehat{B}+3\widehat{B}^{2}\right]\,.$

##  Appendix B Elliptic genera and BPS invariants

In this section, we present the numerator of the elliptic genera for the
following twisted M-strings: $B_{2}$ in table 8, $G_{2}$ in table 9, $B_{3}$
in table 10, $(C_{3})_{0}$ in table 11, and $F_{4}$ in table 12. The elliptic
genera of $(C_{1})_{\pi}$ and $(C_{2})_{\pi}$ theories is listed in table 13.
We also list some refined GV invariants of geometries that engineer 5d
$\mathcal{N}=2$ gauge theory of type $G_{2}$ and $F_{4}$, obtained from the
blowup equations or the elliptic genera, in table 14 and 15.

$\mathfrak{n}$ | $\mathcal{N}$
---|---
[labels*=1,0]B2 | ${1\over 12}(A_{m}B_{+}-A_{+}B_{m})$
[labels*=0,1]B2 | ${1\over 12}(\widehat{A}_{m}\widehat{B}_{+}-\widehat{A}_{+}\widehat{B}_{m})$
[labels*=1,1]B2 | ${1\over 144}(A_{m}B_{-}-A_{-}B_{m})(\widehat{A}_{m}\widehat{B}_{+}-\widehat{A}_{+}\widehat{B}_{m})$
[labels*=2,1]B2 |  $\begin{array}[]{ll}&{1\over 2^{17}3^{9}}(A_{m}B_{+}-A_{+}B_{m})(\widehat{A}_{m}\widehat{B}_{+}-\widehat{A}_{+}\widehat{B}_{m})\Bigr{(}81A_{-}^{2}A_{+}^{4}B_{-}E_{4}^{3}A_{m}-27A_{-}^{3}A_{+}^{4}E_{4}^{3}B_{m}-9A_{+}^{4}B_{-}^{3}\ E_{4}^{2}A_{m}-9A_{-}A_{+}^{4}B_{-}^{2}E_{4}^{2}B_{m}-96A_{-}^{2}A_{+}^{4}B_{-}E_{6}^{2}A_{m}\\\ &+32\ A_{-}^{3}A_{+}^{4}E_{6}^{2}B_{m}-4A_{+}^{4}B_{-}^{3}E_{6}B_{m}-24A_{-}A_{+}^{4}B_{-}^{2}E_{4}\ E_{6}A_{m}-54A_{-}^{3}A_{+}^{3}B_{+}E_{4}^{3}A_{m}-54A_{-}A_{+}^{3}B_{-}^{2}B_{+}E_{4}^{2}\ A_{m}+18A_{-}^{2}A_{+}^{3}B_{-}B_{+}E_{4}^{2}B_{m}+64A_{-}^{3}A_{+}^{3}B_{+}E_{6}^{2}A_{m}-6\ A_{+}^{3}B_{-}^{3}B_{+}E_{4}B_{m}-20A_{+}^{3}B_{-}^{3}B_{+}E_{6}A_{m}\\\ &-12A_{-}A_{+}^{3}B_{-}^{2}\ B_{+}E_{6}B_{m}-24A_{-}^{2}A_{+}^{3}B_{-}B_{+}E_{4}E_{6}A_{m}+24A_{-}^{3}A_{+}^{3}B_{+}E_{4}\ E_{6}B_{m}+36A_{-}^{3}A_{+}^{2}B_{+}^{2}E_{4}^{2}B_{m}-12A_{+}^{2}B_{-}^{3}B_{+}^{2}E_{4}\ A_{m}-36A_{-}A_{+}^{2}B_{-}^{2}B_{+}^{2}E_{6}A_{m}\\\ &+36A_{-}^{2}A_{+}^{2}B_{-}B_{+}^{2}E_{6}\ B_{m}+24A_{-}^{3}A_{+}^{2}B_{+}^{2}E_{4}E_{6}A_{m}+18A_{-}^{3}A_{+}B_{+}^{3}E_{4}^{2}A_{m}-6\ A_{-}A_{+}B_{-}^{2}B_{+}^{3}E_{4}A_{m}+18A_{-}^{2}A_{+}B_{-}B_{+}^{3}E_{4}B_{m}+12A_{-}^{2}\ A_{+}B_{-}B_{+}^{3}E_{6}A_{m}+20A_{-}^{3}A_{+}B_{+}^{3}E_{6}B_{m}\\\ &+3A_{-}^{2}B_{-}B_{+}^{4}\ E_{4}A_{m}+3A_{-}^{3}B_{+}^{4}E_{4}B_{m}+4A_{-}^{3}B_{+}^{4}E_{6}A_{m}+2A_{+}B_{-}^{3}B_{+}^{3}\ B_{m}+B_{-}^{3}B_{+}^{4}A_{m}-3A_{-}B_{-}^{2}B_{+}^{4}B_{m}\Bigr{)}\end{array}$
[labels*=1,2]B2 |  $\begin{array}[]{ll}&{1\over 2^{17}3^{9}}\Bigr{(}-27B_{+}\widehat{A}_{+}^{3}A_{m}\widehat{B}_{m}^{2}\widehat{E}_{4}^{3}\widehat{A}_{-}^{5}+27A_{+}\ \widehat{A}_{+}^{3}B_{m}\widehat{B}_{m}^{2}\widehat{E}_{4}^{3}\widehat{A}_{-}^{5}+81B_{+}\ \widehat{A}_{+}^{2}\widehat{B}_{+}A_{m}\widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{4}^{3}\ \widehat{A}_{-}^{5}-81A_{+}\widehat{A}_{+}^{2}\widehat{B}_{+}B_{m}\widehat{A}_{m}\widehat{B}_{m}\ \widehat{E}_{4}^{3}\widehat{A}_{-}^{5}-9B_{+}\widehat{A}_{+}\widehat{B}_{+}^{2}A_{m}\widehat{B}_{m}^{2}\ \widehat{E}_{4}^{2}\widehat{A}_{-}^{5}+9A_{+}\widehat{A}_{+}\widehat{B}_{+}^{2}B_{m}\widehat{B}_{m}^{2}\ \widehat{E}_{4}^{2}\widehat{A}_{-}^{5}-9B_{+}\widehat{B}_{+}^{3}A_{m}\widehat{A}_{m}\widehat{B}_{m}\ \widehat{E}_{4}^{2}\widehat{A}_{-}^{5}+9A_{+}\widehat{B}_{+}^{3}B_{m}\widehat{A}_{m}\widehat{B}_{m}\ \widehat{E}_{4}^{2}\widehat{A}_{-}^{5}+32B_{+}\widehat{A}_{+}^{3}A_{m}\widehat{B}_{m}^{2}\ \widehat{E}_{6}^{2}\widehat{A}_{-}^{5}\\\ &-32A_{+}\widehat{A}_{+}^{3}B_{m}\widehat{B}_{m}^{2}\ \widehat{E}_{6}^{2}\widehat{A}_{-}^{5}-96B_{+}\widehat{A}_{+}^{2}\widehat{B}_{+}A_{m}\widehat{A}_{m}\ \widehat{B}_{m}\widehat{E}_{6}^{2}\widehat{A}_{-}^{5}+96A_{+}\widehat{A}_{+}^{2}\widehat{B}_{+}B_{m}\ \widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{6}^{2}\widehat{A}_{-}^{5}-4B_{+}\widehat{B}_{+}^{3}A_{m}\ \widehat{B}_{m}^{2}\widehat{E}_{6}\widehat{A}_{-}^{5}+4A_{+}\widehat{B}_{+}^{3}B_{m}\widehat{B}_{m}^{2}\ \widehat{E}_{6}\widehat{A}_{-}^{5}-24B_{+}\widehat{A}_{+}\widehat{B}_{+}^{2}A_{m}\widehat{A}_{m}\ \widehat{B}_{m}\widehat{E}_{4}\widehat{E}_{6}\widehat{A}_{-}^{5}+24A_{+}\widehat{A}_{+}\ \widehat{B}_{+}^{2}B_{m}\widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{4}\widehat{E}_{6}\ \widehat{A}_{-}^{5}-81\widehat{B}_{-}B_{+}\widehat{A}_{+}^{2}\widehat{B}_{+}A_{m}\widehat{A}_{m}^{2}\ \widehat{E}_{4}^{3}\widehat{A}_{-}^{4}+81\widehat{B}_{-}A_{+}\widehat{A}_{+}^{2}\widehat{B}_{+}B_{m}\ \widehat{A}_{m}^{2}\widehat{E}_{4}^{3}\widehat{A}_{-}^{4}\\\ &-27\widehat{B}_{-}B_{+}\widehat{A}_{+}^{3}A_{m}\ \widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{4}^{3}\widehat{A}_{-}^{4}+27\widehat{B}_{-}A_{+}\ \widehat{A}_{+}^{3}B_{m}\widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{4}^{3}\widehat{A}_{-}^{4}+9\ \widehat{B}_{-}B_{+}\widehat{B}_{+}^{3}A_{m}\widehat{A}_{m}^{2}\widehat{E}_{4}^{2}\widehat{A}_{-}^{4}-9\ \widehat{B}_{-}A_{+}\widehat{B}_{+}^{3}B_{m}\widehat{A}_{m}^{2}\widehat{E}_{4}^{2}\widehat{A}_{-}^{4}+18\ \widehat{B}_{-}B_{+}\widehat{A}_{+}^{2}\widehat{B}_{+}A_{m}\widehat{B}_{m}^{2}\widehat{E}_{4}^{2}\ \widehat{A}_{-}^{4}-18\widehat{B}_{-}A_{+}\widehat{A}_{+}^{2}\widehat{B}_{+}B_{m}\widehat{B}_{m}^{2}\ \widehat{E}_{4}^{2}\widehat{A}_{-}^{4}-45\widehat{B}_{-}B_{+}\widehat{A}_{+}\widehat{B}_{+}^{2}A_{m}\ \widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{4}^{2}\widehat{A}_{-}^{4}+45\widehat{B}_{-}A_{+}\ \widehat{A}_{+}\widehat{B}_{+}^{2}B_{m}\widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{4}^{2}\ \widehat{A}_{-}^{4}\\\ &+96\widehat{B}_{-}B_{+}\widehat{A}_{+}^{2}\widehat{B}_{+}A_{m}\widehat{A}_{m}^{2}\ \widehat{E}_{6}^{2}\widehat{A}_{-}^{4}-96\widehat{B}_{-}A_{+}\widehat{A}_{+}^{2}\widehat{B}_{+}B_{m}\ \widehat{A}_{m}^{2}\widehat{E}_{6}^{2}\widehat{A}_{-}^{4}+32\widehat{B}_{-}B_{+}\widehat{A}_{+}^{3}A_{m}\ \widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{6}^{2}\widehat{A}_{-}^{4}-32\widehat{B}_{-}A_{+}\ \widehat{A}_{+}^{3}B_{m}\widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{6}^{2}\widehat{A}_{-}^{4}-6\ \widehat{B}_{-}B_{+}\widehat{B}_{+}^{3}A_{m}\widehat{B}_{m}^{2}\widehat{E}_{4}\widehat{A}_{-}^{4}+6\ \widehat{B}_{-}A_{+}\widehat{B}_{+}^{3}B_{m}\widehat{B}_{m}^{2}\widehat{E}_{4}\widehat{A}_{-}^{4}-12\ \widehat{B}_{-}B_{+}\widehat{A}_{+}\widehat{B}_{+}^{2}A_{m}\widehat{B}_{m}^{2}\widehat{E}_{6}\ \widehat{A}_{-}^{4}+12\widehat{B}_{-}A_{+}\widehat{A}_{+}\widehat{B}_{+}^{2}B_{m}\widehat{B}_{m}^{2}\ \widehat{E}_{6}\widehat{A}_{-}^{4}-16\widehat{B}_{-}B_{+}\widehat{B}_{+}^{3}A_{m}\widehat{A}_{m}\ \widehat{B}_{m}\widehat{E}_{6}\widehat{A}_{-}^{4}\\\ &+16\widehat{B}_{-}A_{+}\widehat{B}_{+}^{3}B_{m}\ \widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{6}\widehat{A}_{-}^{4}+24\widehat{B}_{-}B_{+}\widehat{A}_{+}\ \widehat{B}_{+}^{2}A_{m}\widehat{A}_{m}^{2}\widehat{E}_{4}\widehat{E}_{6}\widehat{A}_{-}^{4}-24\ \widehat{B}_{-}A_{+}\widehat{A}_{+}\widehat{B}_{+}^{2}B_{m}\widehat{A}_{m}^{2}\widehat{E}_{4}\ \widehat{E}_{6}\widehat{A}_{-}^{4}+24\widehat{B}_{-}B_{+}\widehat{A}_{+}^{3}A_{m}\widehat{B}_{m}^{2}\ \widehat{E}_{4}\widehat{E}_{6}\widehat{A}_{-}^{4}-24\widehat{B}_{-}A_{+}\widehat{A}_{+}^{3}B_{m}\ \widehat{B}_{m}^{2}\widehat{E}_{4}\widehat{E}_{6}\widehat{A}_{-}^{4}-24\widehat{B}_{-}B_{+}\ \widehat{A}_{+}^{2}\widehat{B}_{+}A_{m}\widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{4}\widehat{E}_{6}\ \widehat{A}_{-}^{4}+24\widehat{B}_{-}A_{+}\widehat{A}_{+}^{2}\widehat{B}_{+}B_{m}\widehat{A}_{m}\ \widehat{B}_{m}\widehat{E}_{4}\widehat{E}_{6}\widehat{A}_{-}^{4}+54\widehat{B}_{-}^{2}B_{+}\ \widehat{A}_{+}^{3}A_{m}\widehat{A}_{m}^{2}\widehat{E}_{4}^{3}\widehat{A}_{-}^{3}\\\ &+54B_{-}\ \widehat{A}_{+}^{3}\widehat{B}_{+}^{2}A_{m}\widehat{A}_{m}^{2}\widehat{E}_{4}^{3}\widehat{A}_{-}^{3}-54\ \widehat{B}_{-}^{2}A_{+}\widehat{A}_{+}^{3}B_{m}\widehat{A}_{m}^{2}\widehat{E}_{4}^{3}\ \widehat{A}_{-}^{3}-54A_{-}\widehat{A}_{+}^{3}\widehat{B}_{+}^{2}B_{m}\widehat{A}_{m}^{2}\ \widehat{E}_{4}^{3}\widehat{A}_{-}^{3}+54B_{-}\widehat{A}_{+}^{5}A_{m}\widehat{B}_{m}^{2}\ \widehat{E}_{4}^{3}\widehat{A}_{-}^{3}-54A_{-}\widehat{A}_{+}^{5}B_{m}\widehat{B}_{m}^{2}\ \widehat{E}_{4}^{3}\widehat{A}_{-}^{3}-108B_{-}\widehat{A}_{+}^{4}\widehat{B}_{+}A_{m}\widehat{A}_{m}\ \widehat{B}_{m}\widehat{E}_{4}^{3}\widehat{A}_{-}^{3}+108A_{-}\widehat{A}_{+}^{4}\widehat{B}_{+}B_{m}\ \widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{4}^{3}\widehat{A}_{-}^{3}-18B_{-}\widehat{A}_{+}\ \widehat{B}_{+}^{4}A_{m}\widehat{A}_{m}^{2}\widehat{E}_{4}^{2}\widehat{A}_{-}^{3}+54\widehat{B}_{-}^{2}\ B_{+}\widehat{A}_{+}\widehat{B}_{+}^{2}A_{m}\widehat{A}_{m}^{2}\widehat{E}_{4}^{2}\widehat{A}_{-}^{3}\\\ &+18\ A_{-}\widehat{A}_{+}\widehat{B}_{+}^{4}B_{m}\widehat{A}_{m}^{2}\widehat{E}_{4}^{2}\widehat{A}_{-}^{3}-54\ \widehat{B}_{-}^{2}A_{+}\widehat{A}_{+}\widehat{B}_{+}^{2}B_{m}\widehat{A}_{m}^{2}\widehat{E}_{4}^{2}\ \widehat{A}_{-}^{3}+36\widehat{B}_{-}^{2}B_{+}\widehat{A}_{+}^{3}A_{m}\widehat{B}_{m}^{2}\ \widehat{E}_{4}^{2}\widehat{A}_{-}^{3}-18B_{-}\widehat{A}_{+}^{3}\widehat{B}_{+}^{2}A_{m}\ \widehat{B}_{m}^{2}\widehat{E}_{4}^{2}\widehat{A}_{-}^{3}-36\widehat{B}_{-}^{2}A_{+}\widehat{A}_{+}^{3}\ B_{m}\widehat{B}_{m}^{2}\widehat{E}_{4}^{2}\widehat{A}_{-}^{3}+18A_{-}\widehat{A}_{+}^{3}\ \widehat{B}_{+}^{2}B_{m}\widehat{B}_{m}^{2}\widehat{E}_{4}^{2}\widehat{A}_{-}^{3}+36B_{-}\ \widehat{A}_{+}^{2}\widehat{B}_{+}^{3}A_{m}\widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{4}^{2}\ \widehat{A}_{-}^{3}-18\widehat{B}_{-}^{2}B_{+}\widehat{A}_{+}^{2}\widehat{B}_{+}A_{m}\widehat{A}_{m}\ \widehat{B}_{m}\widehat{E}_{4}^{2}\widehat{A}_{-}^{3}\\\ &-36A_{-}\widehat{A}_{+}^{2}\widehat{B}_{+}^{3}B_{m}\ \widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{4}^{2}\widehat{A}_{-}^{3}+18\widehat{B}_{-}^{2}A_{+}\ \widehat{A}_{+}^{2}\widehat{B}_{+}B_{m}\widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{4}^{2}\ \widehat{A}_{-}^{3}-64\widehat{B}_{-}^{2}B_{+}\widehat{A}_{+}^{3}A_{m}\widehat{A}_{m}^{2}\ \widehat{E}_{6}^{2}\widehat{A}_{-}^{3}-64B_{-}\widehat{A}_{+}^{3}\widehat{B}_{+}^{2}A_{m}\ \widehat{A}_{m}^{2}\widehat{E}_{6}^{2}\widehat{A}_{-}^{3}+64\widehat{B}_{-}^{2}A_{+}\widehat{A}_{+}^{3}\ B_{m}\widehat{A}_{m}^{2}\widehat{E}_{6}^{2}\widehat{A}_{-}^{3}+64A_{-}\widehat{A}_{+}^{3}\ \widehat{B}_{+}^{2}B_{m}\widehat{A}_{m}^{2}\widehat{E}_{6}^{2}\widehat{A}_{-}^{3}-64B_{-}\ \widehat{A}_{+}^{5}A_{m}\widehat{B}_{m}^{2}\widehat{E}_{6}^{2}\widehat{A}_{-}^{3}+64A_{-}\ \widehat{A}_{+}^{5}B_{m}\widehat{B}_{m}^{2}\widehat{E}_{6}^{2}\widehat{A}_{-}^{3}+128B_{-}\ \widehat{A}_{+}^{4}\widehat{B}_{+}A_{m}\widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{6}^{2}\ \widehat{A}_{-}^{3}\\\ &-128A_{-}\widehat{A}_{+}^{4}\widehat{B}_{+}B_{m}\widehat{A}_{m}\widehat{B}_{m}\ \widehat{E}_{6}^{2}\widehat{A}_{-}^{3}-6\widehat{B}_{-}^{2}B_{+}\widehat{B}_{+}^{3}A_{m}\widehat{A}_{m}\ \widehat{B}_{m}\widehat{E}_{4}\widehat{A}_{-}^{3}+6\widehat{B}_{-}^{2}A_{+}\widehat{B}_{+}^{3}B_{m}\ \widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{4}\widehat{A}_{-}^{3}-4B_{-}\widehat{B}_{+}^{5}A_{m}\ \widehat{A}_{m}^{2}\widehat{E}_{6}\widehat{A}_{-}^{3}+20\widehat{B}_{-}^{2}B_{+}\widehat{B}_{+}^{3}A_{m}\ \widehat{A}_{m}^{2}\widehat{E}_{6}\widehat{A}_{-}^{3}+4A_{-}\widehat{B}_{+}^{5}B_{m}\widehat{A}_{m}^{2}\ \widehat{E}_{6}\widehat{A}_{-}^{3}-20\widehat{B}_{-}^{2}A_{+}\widehat{B}_{+}^{3}B_{m}\widehat{A}_{m}^{2}\ \widehat{E}_{6}\widehat{A}_{-}^{3}-4B_{-}\widehat{A}_{+}^{2}\widehat{B}_{+}^{3}A_{m}\widehat{B}_{m}^{2}\ \widehat{E}_{6}\widehat{A}_{-}^{3}+36\widehat{B}_{-}^{2}B_{+}\widehat{A}_{+}^{2}\widehat{B}_{+}A_{m}\ \widehat{B}_{m}^{2}\widehat{E}_{6}\widehat{A}_{-}^{3}+4A_{-}\widehat{A}_{+}^{2}\widehat{B}_{+}^{3}B_{m}\ \widehat{B}_{m}^{2}\widehat{E}_{6}\widehat{A}_{-}^{3}\\\ &-36\widehat{B}_{-}^{2}A_{+}\widehat{A}_{+}^{2}\ \widehat{B}_{+}B_{m}\widehat{B}_{m}^{2}\widehat{E}_{6}\widehat{A}_{-}^{3}+8B_{-}\widehat{A}_{+}\ \widehat{B}_{+}^{4}A_{m}\widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{6}\widehat{A}_{-}^{3}-24\ \widehat{B}_{-}^{2}B_{+}\widehat{A}_{+}\widehat{B}_{+}^{2}A_{m}\widehat{A}_{m}\widehat{B}_{m}\ \widehat{E}_{6}\widehat{A}_{-}^{3}-8A_{-}\widehat{A}_{+}\widehat{B}_{+}^{4}B_{m}\widehat{A}_{m}\ \widehat{B}_{m}\widehat{E}_{6}\widehat{A}_{-}^{3}+24\widehat{B}_{-}^{2}A_{+}\widehat{A}_{+}\ \widehat{B}_{+}^{2}B_{m}\widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{6}\widehat{A}_{-}^{3}-24B_{-}\ \widehat{A}_{+}^{2}\widehat{B}_{+}^{3}A_{m}\widehat{A}_{m}^{2}\widehat{E}_{4}\widehat{E}_{6}\ \widehat{A}_{-}^{3}+24\widehat{B}_{-}^{2}B_{+}\widehat{A}_{+}^{2}\widehat{B}_{+}A_{m}\widehat{A}_{m}^{2}\ \widehat{E}_{4}\widehat{E}_{6}\widehat{A}_{-}^{3}+24A_{-}\widehat{A}_{+}^{2}\widehat{B}_{+}^{3}B_{m}\ \widehat{A}_{m}^{2}\widehat{E}_{4}\widehat{E}_{6}\widehat{A}_{-}^{3}\\\ &-24\widehat{B}_{-}^{2}A_{+}\ \widehat{A}_{+}^{2}\widehat{B}_{+}B_{m}\widehat{A}_{m}^{2}\widehat{E}_{4}\widehat{E}_{6}\ \widehat{A}_{-}^{3}-24B_{-}\widehat{A}_{+}^{4}\widehat{B}_{+}A_{m}\widehat{B}_{m}^{2}\widehat{E}_{4}\ \widehat{E}_{6}\widehat{A}_{-}^{3}+24A_{-}\widehat{A}_{+}^{4}\widehat{B}_{+}B_{m}\widehat{B}_{m}^{2}\ \widehat{E}_{4}\widehat{E}_{6}\widehat{A}_{-}^{3}+48B_{-}\widehat{A}_{+}^{3}\widehat{B}_{+}^{2}A_{m}\ \widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{4}\widehat{E}_{6}\widehat{A}_{-}^{3}-48A_{-}\ \widehat{A}_{+}^{3}\widehat{B}_{+}^{2}B_{m}\widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{4}\widehat{E}_{6}\ \widehat{A}_{-}^{3}+2\widehat{B}_{-}^{3}B_{+}\widehat{B}_{+}^{3}A_{m}\widehat{B}_{m}^{2}\ \widehat{A}_{-}^{2}-2\widehat{B}_{-}^{3}A_{+}\widehat{B}_{+}^{3}B_{m}\widehat{B}_{m}^{2}\ \widehat{A}_{-}^{2}-54B_{-}\widehat{B}_{-}\widehat{A}_{+}^{2}\widehat{B}_{+}^{3}A_{m}\widehat{A}_{m}^{2}\ \widehat{E}_{4}^{2}\widehat{A}_{-}^{2}\\\ &+54A_{-}\widehat{B}_{-}\widehat{A}_{+}^{2}\widehat{B}_{+}^{3}B_{m}\ \widehat{A}_{m}^{2}\widehat{E}_{4}^{2}\widehat{A}_{-}^{2}-54B_{-}\widehat{B}_{-}\widehat{A}_{+}^{4}\ \widehat{B}_{+}A_{m}\widehat{B}_{m}^{2}\widehat{E}_{4}^{2}\widehat{A}_{-}^{2}+54A_{-}\widehat{B}_{-}\ \widehat{A}_{+}^{4}\widehat{B}_{+}B_{m}\widehat{B}_{m}^{2}\widehat{E}_{4}^{2}\widehat{A}_{-}^{2}-18\ \widehat{B}_{-}^{3}B_{+}\widehat{A}_{+}^{3}A_{m}\widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{4}^{2}\ \widehat{A}_{-}^{2}+108B_{-}\widehat{B}_{-}\widehat{A}_{+}^{3}\widehat{B}_{+}^{2}A_{m}\widehat{A}_{m}\ \widehat{B}_{m}\widehat{E}_{4}^{2}\widehat{A}_{-}^{2}+18\widehat{B}_{-}^{3}A_{+}\widehat{A}_{+}^{3}B_{m}\ \widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{4}^{2}\widehat{A}_{-}^{2}-108A_{-}\widehat{B}_{-}\ \widehat{A}_{+}^{3}\widehat{B}_{+}^{2}B_{m}\widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{4}^{2}\ \widehat{A}_{-}^{2}-6B_{-}\widehat{B}_{-}\widehat{B}_{+}^{5}A_{m}\widehat{A}_{m}^{2}\widehat{E}_{4}\ \widehat{A}_{-}^{2}\\\ &+12\widehat{B}_{-}^{3}B_{+}\widehat{B}_{+}^{3}A_{m}\widehat{A}_{m}^{2}\widehat{E}_{4}\ \widehat{A}_{-}^{2}+6A_{-}\widehat{B}_{-}\widehat{B}_{+}^{5}B_{m}\widehat{A}_{m}^{2}\widehat{E}_{4}\ \widehat{A}_{-}^{2}-12\widehat{B}_{-}^{3}A_{+}\widehat{B}_{+}^{3}B_{m}\widehat{A}_{m}^{2}\widehat{E}_{4}\ \widehat{A}_{-}^{2}-6B_{-}\widehat{B}_{-}\widehat{A}_{+}^{2}\widehat{B}_{+}^{3}A_{m}\widehat{B}_{m}^{2}\ \widehat{E}_{4}\widehat{A}_{-}^{2}+18\widehat{B}_{-}^{3}B_{+}\widehat{A}_{+}^{2}\widehat{B}_{+}A_{m}\ \widehat{B}_{m}^{2}\widehat{E}_{4}\widehat{A}_{-}^{2}+6A_{-}\widehat{B}_{-}\widehat{A}_{+}^{2}\ \widehat{B}_{+}^{3}B_{m}\widehat{B}_{m}^{2}\widehat{E}_{4}\widehat{A}_{-}^{2}-18\widehat{B}_{-}^{3}A_{+}\ \widehat{A}_{+}^{2}\widehat{B}_{+}B_{m}\widehat{B}_{m}^{2}\widehat{E}_{4}\widehat{A}_{-}^{2}+12B_{-}\ \widehat{B}_{-}\widehat{A}_{+}\widehat{B}_{+}^{4}A_{m}\widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{4}\ \widehat{A}_{-}^{2}-6\widehat{B}_{-}^{3}B_{+}\widehat{A}_{+}\widehat{B}_{+}^{2}A_{m}\widehat{A}_{m}\ \widehat{B}_{m}\widehat{E}_{4}\widehat{A}_{-}^{2}\\\ &-12A_{-}\widehat{B}_{-}\widehat{A}_{+}\ \widehat{B}_{+}^{4}B_{m}\widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{4}\widehat{A}_{-}^{2}+6\ \widehat{B}_{-}^{3}A_{+}\widehat{A}_{+}\widehat{B}_{+}^{2}B_{m}\widehat{A}_{m}\widehat{B}_{m}\ \widehat{E}_{4}\widehat{A}_{-}^{2}-36B_{-}\widehat{B}_{-}\widehat{A}_{+}\widehat{B}_{+}^{4}A_{m}\ \widehat{A}_{m}^{2}\widehat{E}_{6}\widehat{A}_{-}^{2}+36\widehat{B}_{-}^{3}B_{+}\widehat{A}_{+}\ \widehat{B}_{+}^{2}A_{m}\widehat{A}_{m}^{2}\widehat{E}_{6}\widehat{A}_{-}^{2}+36A_{-}\widehat{B}_{-}\ \widehat{A}_{+}\widehat{B}_{+}^{4}B_{m}\widehat{A}_{m}^{2}\widehat{E}_{6}\widehat{A}_{-}^{2}-36\ \widehat{B}_{-}^{3}A_{+}\widehat{A}_{+}\widehat{B}_{+}^{2}B_{m}\widehat{A}_{m}^{2}\widehat{E}_{6}\ \widehat{A}_{-}^{2}+20\widehat{B}_{-}^{3}B_{+}\widehat{A}_{+}^{3}A_{m}\widehat{B}_{m}^{2}\widehat{E}_{6}\ \widehat{A}_{-}^{2}-36B_{-}\widehat{B}_{-}\widehat{A}_{+}^{3}\widehat{B}_{+}^{2}A_{m}\widehat{B}_{m}^{2}\ \widehat{E}_{6}\widehat{A}_{-}^{2}\\\ &-20\widehat{B}_{-}^{3}A_{+}\widehat{A}_{+}^{3}B_{m}\widehat{B}_{m}^{2}\ \widehat{E}_{6}\widehat{A}_{-}^{2}+36A_{-}\widehat{B}_{-}\widehat{A}_{+}^{3}\widehat{B}_{+}^{2}B_{m}\ \widehat{B}_{m}^{2}\widehat{E}_{6}\widehat{A}_{-}^{2}+72B_{-}\widehat{B}_{-}\widehat{A}_{+}^{2}\ \widehat{B}_{+}^{3}A_{m}\widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{6}\widehat{A}_{-}^{2}-24\ \widehat{B}_{-}^{3}B_{+}\widehat{A}_{+}^{2}\widehat{B}_{+}A_{m}\widehat{A}_{m}\widehat{B}_{m}\ \widehat{E}_{6}\widehat{A}_{-}^{2}-72A_{-}\widehat{B}_{-}\widehat{A}_{+}^{2}\widehat{B}_{+}^{3}B_{m}\ \widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{6}\widehat{A}_{-}^{2}+24\widehat{B}_{-}^{3}A_{+}\ \widehat{A}_{+}^{2}\widehat{B}_{+}B_{m}\widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{6}\ \widehat{A}_{-}^{2}-24\widehat{B}_{-}^{3}B_{+}\widehat{A}_{+}^{3}A_{m}\widehat{A}_{m}^{2}\widehat{E}_{4}\ \widehat{E}_{6}\widehat{A}_{-}^{2}-24B_{-}\widehat{B}_{-}\widehat{A}_{+}^{3}\widehat{B}_{+}^{2}A_{m}\ \widehat{A}_{m}^{2}\widehat{E}_{4}\widehat{E}_{6}\widehat{A}_{-}^{2}\\\ &+24\widehat{B}_{-}^{3}A_{+}\ \widehat{A}_{+}^{3}B_{m}\widehat{A}_{m}^{2}\widehat{E}_{4}\widehat{E}_{6}\widehat{A}_{-}^{2}+24A_{-}\ \widehat{B}_{-}\widehat{A}_{+}^{3}\widehat{B}_{+}^{2}B_{m}\widehat{A}_{m}^{2}\widehat{E}_{4}\widehat{E}_{6}\ \widehat{A}_{-}^{2}-24B_{-}\widehat{B}_{-}\widehat{A}_{+}^{5}A_{m}\widehat{B}_{m}^{2}\widehat{E}_{4}\ \widehat{E}_{6}\widehat{A}_{-}^{2}+24A_{-}\widehat{B}_{-}\widehat{A}_{+}^{5}B_{m}\widehat{B}_{m}^{2}\ \widehat{E}_{4}\widehat{E}_{6}\widehat{A}_{-}^{2}+48B_{-}\widehat{B}_{-}\widehat{A}_{+}^{4}\ \widehat{B}_{+}A_{m}\widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{4}\widehat{E}_{6}\widehat{A}_{-}^{2}-48\ A_{-}\widehat{B}_{-}\widehat{A}_{+}^{4}\widehat{B}_{+}B_{m}\widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{4}\ \widehat{E}_{6}\widehat{A}_{-}^{2}-3\widehat{B}_{-}^{4}B_{+}\widehat{A}_{+}\widehat{B}_{+}^{2}A_{m}\ \widehat{B}_{m}^{2}\widehat{A}_{-}+3\widehat{B}_{-}^{4}A_{+}\widehat{A}_{+}\widehat{B}_{+}^{2}B_{m}\ \widehat{B}_{m}^{2}\widehat{A}_{-}\\\ &-18\widehat{B}_{-}^{4}B_{+}\widehat{A}_{+}^{3}A_{m}\widehat{A}_{m}^{2}\ \widehat{E}_{4}^{2}\widehat{A}_{-}-18B_{-}\widehat{B}_{-}^{2}\widehat{A}_{+}^{3}\widehat{B}_{+}^{2}A_{m}\ \widehat{A}_{m}^{2}\widehat{E}_{4}^{2}\widehat{A}_{-}+18\widehat{B}_{-}^{4}A_{+}\widehat{A}_{+}^{3}B_{m}\ \widehat{A}_{m}^{2}\widehat{E}_{4}^{2}\widehat{A}_{-}+18A_{-}\widehat{B}_{-}^{2}\widehat{A}_{+}^{3}\ \widehat{B}_{+}^{2}B_{m}\widehat{A}_{m}^{2}\widehat{E}_{4}^{2}\widehat{A}_{-}-18B_{-}\widehat{B}_{-}^{2}\ \widehat{A}_{+}^{5}A_{m}\widehat{B}_{m}^{2}\widehat{E}_{4}^{2}\widehat{A}_{-}+18A_{-}\widehat{B}_{-}^{2}\ \widehat{A}_{+}^{5}B_{m}\widehat{B}_{m}^{2}\widehat{E}_{4}^{2}\widehat{A}_{-}+36B_{-}\widehat{B}_{-}^{2}\ \widehat{A}_{+}^{4}\widehat{B}_{+}A_{m}\widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{4}^{2}\ \widehat{A}_{-}-36A_{-}\widehat{B}_{-}^{2}\widehat{A}_{+}^{4}\widehat{B}_{+}B_{m}\widehat{A}_{m}\ \widehat{B}_{m}\widehat{E}_{4}^{2}\widehat{A}_{-}\\\ &-\widehat{B}_{-}^{4}B_{+}\widehat{B}_{+}^{3}A_{m}\ \widehat{A}_{m}\widehat{B}_{m}\widehat{A}_{-}+\widehat{B}_{-}^{4}A_{+}\widehat{B}_{+}^{3}B_{m}\ \widehat{A}_{m}\widehat{B}_{m}\widehat{A}_{-}-18B_{-}\widehat{B}_{-}^{2}\widehat{A}_{+}\ \widehat{B}_{+}^{4}A_{m}\widehat{A}_{m}^{2}\widehat{E}_{4}\widehat{A}_{-}+6\widehat{B}_{-}^{4}B_{+}\ \widehat{A}_{+}\widehat{B}_{+}^{2}A_{m}\widehat{A}_{m}^{2}\widehat{E}_{4}\widehat{A}_{-}+18A_{-}\ \widehat{B}_{-}^{2}\widehat{A}_{+}\widehat{B}_{+}^{4}B_{m}\widehat{A}_{m}^{2}\widehat{E}_{4}\ \widehat{A}_{-}-6\widehat{B}_{-}^{4}A_{+}\widehat{A}_{+}\widehat{B}_{+}^{2}B_{m}\widehat{A}_{m}^{2}\ \widehat{E}_{4}\widehat{A}_{-}+3\widehat{B}_{-}^{4}B_{+}\widehat{A}_{+}^{3}A_{m}\widehat{B}_{m}^{2}\ \widehat{E}_{4}\widehat{A}_{-}-18B_{-}\widehat{B}_{-}^{2}\widehat{A}_{+}^{3}\widehat{B}_{+}^{2}A_{m}\ \widehat{B}_{m}^{2}\widehat{E}_{4}\widehat{A}_{-}-3\widehat{B}_{-}^{4}A_{+}\widehat{A}_{+}^{3}B_{m}\ \widehat{B}_{m}^{2}\widehat{E}_{4}\widehat{A}_{-}\\\ &+18A_{-}\widehat{B}_{-}^{2}\widehat{A}_{+}^{3}\ \widehat{B}_{+}^{2}B_{m}\widehat{B}_{m}^{2}\widehat{E}_{4}\widehat{A}_{-}+36B_{-}\widehat{B}_{-}^{2}\ \widehat{A}_{+}^{2}\widehat{B}_{+}^{3}A_{m}\widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{4}\ \widehat{A}_{-}-15\widehat{B}_{-}^{4}B_{+}\widehat{A}_{+}^{2}\widehat{B}_{+}A_{m}\widehat{A}_{m}\ \widehat{B}_{m}\widehat{E}_{4}\widehat{A}_{-}-36A_{-}\widehat{B}_{-}^{2}\widehat{A}_{+}^{2}\ \widehat{B}_{+}^{3}B_{m}\widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{4}\widehat{A}_{-}+15\ \widehat{B}_{-}^{4}A_{+}\widehat{A}_{+}^{2}\widehat{B}_{+}B_{m}\widehat{A}_{m}\widehat{B}_{m}\ \widehat{E}_{4}\widehat{A}_{-}-36B_{-}\widehat{B}_{-}^{2}\widehat{A}_{+}^{2}\widehat{B}_{+}^{3}A_{m}\ \widehat{A}_{m}^{2}\widehat{E}_{6}\widehat{A}_{-}-12\widehat{B}_{-}^{4}B_{+}\widehat{A}_{+}^{2}\ \widehat{B}_{+}A_{m}\widehat{A}_{m}^{2}\widehat{E}_{6}\widehat{A}_{-}+36A_{-}\widehat{B}_{-}^{2}\ \widehat{A}_{+}^{2}\widehat{B}_{+}^{3}B_{m}\widehat{A}_{m}^{2}\widehat{E}_{6}\widehat{A}_{-}\\\ &+12\ \widehat{B}_{-}^{4}A_{+}\widehat{A}_{+}^{2}\widehat{B}_{+}B_{m}\widehat{A}_{m}^{2}\widehat{E}_{6}\ \widehat{A}_{-}-36B_{-}\widehat{B}_{-}^{2}\widehat{A}_{+}^{4}\widehat{B}_{+}A_{m}\widehat{B}_{m}^{2}\ \widehat{E}_{6}\widehat{A}_{-}+36A_{-}\widehat{B}_{-}^{2}\widehat{A}_{+}^{4}\widehat{B}_{+}B_{m}\ \widehat{B}_{m}^{2}\widehat{E}_{6}\widehat{A}_{-}-16\widehat{B}_{-}^{4}B_{+}\widehat{A}_{+}^{3}A_{m}\ \widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{6}\widehat{A}_{-}+72B_{-}\widehat{B}_{-}^{2}\ \widehat{A}_{+}^{3}\widehat{B}_{+}^{2}A_{m}\widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{6}\ \widehat{A}_{-}+16\widehat{B}_{-}^{4}A_{+}\widehat{A}_{+}^{3}B_{m}\widehat{A}_{m}\widehat{B}_{m}\ \widehat{E}_{6}\widehat{A}_{-}-72A_{-}\widehat{B}_{-}^{2}\widehat{A}_{+}^{3}\widehat{B}_{+}^{2}B_{m}\ \widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{6}\widehat{A}_{-}+2B_{-}\widehat{B}_{-}^{3}\widehat{B}_{+}^{5}\ A_{m}\widehat{A}_{m}^{2}-\widehat{B}_{-}^{5}B_{+}\widehat{B}_{+}^{3}A_{m}\widehat{A}_{m}^{2}\\\ &-2A_{-}\ \widehat{B}_{-}^{3}\widehat{B}_{+}^{5}B_{m}\widehat{A}_{m}^{2}+\widehat{B}_{-}^{5}A_{+}\widehat{B}_{+}^{3}\ B_{m}\widehat{A}_{m}^{2}+2B_{-}\widehat{B}_{-}^{3}\widehat{A}_{+}^{2}\widehat{B}_{+}^{3}A_{m}\ \widehat{B}_{m}^{2}-2A_{-}\widehat{B}_{-}^{3}\widehat{A}_{+}^{2}\widehat{B}_{+}^{3}B_{m}\ \widehat{B}_{m}^{2}-4B_{-}\widehat{B}_{-}^{3}\widehat{A}_{+}\widehat{B}_{+}^{4}A_{m}\widehat{A}_{m}\ \widehat{B}_{m}+3\widehat{B}_{-}^{5}B_{+}\widehat{A}_{+}\widehat{B}_{+}^{2}A_{m}\widehat{A}_{m}\ \widehat{B}_{m}+4A_{-}\widehat{B}_{-}^{3}\widehat{A}_{+}\widehat{B}_{+}^{4}B_{m}\widehat{A}_{m}\ \widehat{B}_{m}-3\widehat{B}_{-}^{5}A_{+}\widehat{A}_{+}\widehat{B}_{+}^{2}B_{m}\widehat{A}_{m}\ \widehat{B}_{m}-6B_{-}\widehat{B}_{-}^{3}\widehat{A}_{+}^{2}\widehat{B}_{+}^{3}A_{m}\widehat{A}_{m}^{2}\ \widehat{E}_{4}-3\widehat{B}_{-}^{5}B_{+}\widehat{A}_{+}^{2}\widehat{B}_{+}A_{m}\widehat{A}_{m}^{2}\ \widehat{E}_{4}\\\ &+6A_{-}\widehat{B}_{-}^{3}\widehat{A}_{+}^{2}\widehat{B}_{+}^{3}B_{m}\widehat{A}_{m}^{2}\ \widehat{E}_{4}+3\widehat{B}_{-}^{5}A_{+}\widehat{A}_{+}^{2}\widehat{B}_{+}B_{m}\widehat{A}_{m}^{2}\ \widehat{E}_{4}-6B_{-}\widehat{B}_{-}^{3}\widehat{A}_{+}^{4}\widehat{B}_{+}A_{m}\widehat{B}_{m}^{2}\ \widehat{E}_{4}+6A_{-}\widehat{B}_{-}^{3}\widehat{A}_{+}^{4}\widehat{B}_{+}B_{m}\widehat{B}_{m}^{2}\ \widehat{E}_{4}-3\widehat{B}_{-}^{5}B_{+}\widehat{A}_{+}^{3}A_{m}\widehat{A}_{m}\widehat{B}_{m}\ \widehat{E}_{4}+12B_{-}\widehat{B}_{-}^{3}\widehat{A}_{+}^{3}\widehat{B}_{+}^{2}A_{m}\widehat{A}_{m}\ \widehat{B}_{m}\widehat{E}_{4}+3\widehat{B}_{-}^{5}A_{+}\widehat{A}_{+}^{3}B_{m}\widehat{A}_{m}\ \widehat{B}_{m}\widehat{E}_{4}-12A_{-}\widehat{B}_{-}^{3}\widehat{A}_{+}^{3}\widehat{B}_{+}^{2}B_{m}\ \widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{4}-4\widehat{B}_{-}^{5}B_{+}\widehat{A}_{+}^{3}A_{m}\ \widehat{A}_{m}^{2}\widehat{E}_{6}-4B_{-}\widehat{B}_{-}^{3}\widehat{A}_{+}^{3}\widehat{B}_{+}^{2}A_{m}\ \widehat{A}_{m}^{2}\widehat{E}_{6}\\\ &+4\widehat{B}_{-}^{5}A_{+}\widehat{A}_{+}^{3}B_{m}\widehat{A}_{m}^{2}\ \widehat{E}_{6}+4A_{-}\widehat{B}_{-}^{3}\widehat{A}_{+}^{3}\widehat{B}_{+}^{2}B_{m}\widehat{A}_{m}^{2}\ \widehat{E}_{6}-4B_{-}\widehat{B}_{-}^{3}\widehat{A}_{+}^{5}A_{m}\widehat{B}_{m}^{2}\widehat{E}_{6}+4\ A_{-}\widehat{B}_{-}^{3}\widehat{A}_{+}^{5}B_{m}\widehat{B}_{m}^{2}\widehat{E}_{6}+8B_{-}\ \widehat{B}_{-}^{3}\widehat{A}_{+}^{4}\widehat{B}_{+}A_{m}\widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{6}-8\ A_{-}\widehat{B}_{-}^{3}\widehat{A}_{+}^{4}\widehat{B}_{+}B_{m}\widehat{A}_{m}\widehat{B}_{m}\widehat{E}_{6}\Bigr{)}\end{array}$
Table 8: Elliptic genera of $B_{2}$. Here, $A_{z}=A_{z}(\tau)$, $\widehat{A}_{z}=A_{z}(2\tau)$, and same for $B_{z}$. $\mathfrak{n}$ | $\mathcal{N}$
---|---
[reverse arrows, labels*=0,1]G2 | ${1\over 12}(A_{m}B_{+}-A_{+}B_{m})$
[reverse arrows, labels*=1,0]G2 | ${1\over 12}(\widehat{A}_{m}\widehat{B}_{+}-\widehat{A}_{+}\widehat{B}_{m})$
[reverse arrows, labels*=1,1]G2 | ${1\over 144}(A_{m}B_{-}-A_{-}B_{m})(\widehat{A}_{m}\widehat{B}_{+}-\widehat{A}_{+}\widehat{B}_{m})$
[reverse arrows, labels*=2,1]G2 |  $\begin{array}[]{ll}&{1\over 2^{17}3^{9}}(\widehat{A}_{m}\widehat{B}_{-}-\widehat{A}_{-}\widehat{B}_{m})(\widehat{A}_{m}\widehat{B}_{+}-\widehat{A}_{+}\widehat{B}_{m})\Bigr{(}-27B_{+}A_{m}\widehat{E}_{4}^{3}\widehat{A}_{-}^{6}+27A_{+}B_{m}\widehat{E}_{4}^{3}\ \widehat{A}_{-}^{6}+32B_{+}A_{m}\widehat{E}_{6}^{2}\widehat{A}_{-}^{6}-32A_{+}B_{m}\widehat{E}_{6}^{2}\ \widehat{A}_{-}^{6}\\\ &+24\widehat{B}_{-}B_{+}A_{m}\widehat{E}_{4}\widehat{E}_{6}\widehat{A}_{-}^{5}-24\ \widehat{B}_{-}A_{+}B_{m}\widehat{E}_{4}\widehat{E}_{6}\widehat{A}_{-}^{5}+45\widehat{B}_{-}^{2}B_{+}\ A_{m}\widehat{E}_{4}^{2}\widehat{A}_{-}^{4}-45\widehat{B}_{-}^{2}A_{+}B_{m}\widehat{E}_{4}^{2}\ \widehat{A}_{-}^{4}+54B_{-}\widehat{A}_{+}^{3}A_{m}\widehat{E}_{4}^{3}\widehat{A}_{-}^{3}-54A_{-}\ \widehat{A}_{+}^{3}B_{m}\widehat{E}_{4}^{3}\widehat{A}_{-}^{3}\\\ &-18B_{-}\widehat{A}_{+}\widehat{B}_{+}^{2}\ A_{m}\widehat{E}_{4}^{2}\widehat{A}_{-}^{3}+18A_{-}\widehat{A}_{+}\widehat{B}_{+}^{2}B_{m}\ \widehat{E}_{4}^{2}\widehat{A}_{-}^{3}-64B_{-}\widehat{A}_{+}^{3}A_{m}\widehat{E}_{6}^{2}\ \widehat{A}_{-}^{3}+64A_{-}\widehat{A}_{+}^{3}B_{m}\widehat{E}_{6}^{2}\widehat{A}_{-}^{3}-4B_{-}\ \widehat{B}_{+}^{3}A_{m}\widehat{E}_{6}\widehat{A}_{-}^{3}+40\widehat{B}_{-}^{3}B_{+}A_{m}\ \widehat{E}_{6}\widehat{A}_{-}^{3}\\\ &+4A_{-}\widehat{B}_{+}^{3}B_{m}\widehat{E}_{6}\widehat{A}_{-}^{3}-40\ \widehat{B}_{-}^{3}A_{+}B_{m}\widehat{E}_{6}\widehat{A}_{-}^{3}-24B_{-}\widehat{A}_{+}^{2}\ \widehat{B}_{+}A_{m}\widehat{E}_{4}\widehat{E}_{6}\widehat{A}_{-}^{3}+24A_{-}\widehat{A}_{+}^{2}\ \widehat{B}_{+}B_{m}\widehat{E}_{4}\widehat{E}_{6}\widehat{A}_{-}^{3}-54B_{-}\widehat{B}_{-}\ \widehat{A}_{+}^{2}\widehat{B}_{+}A_{m}\widehat{E}_{4}^{2}\widehat{A}_{-}^{2}+54A_{-}\widehat{B}_{-}\ \widehat{A}_{+}^{2}\widehat{B}_{+}B_{m}\widehat{E}_{4}^{2}\widehat{A}_{-}^{2}-6B_{-}\widehat{B}_{-}\ \widehat{B}_{+}^{3}A_{m}\widehat{E}_{4}\widehat{A}_{-}^{2}\\\ &+15\widehat{B}_{-}^{4}B_{+}A_{m}\ \widehat{E}_{4}\widehat{A}_{-}^{2}+6A_{-}\widehat{B}_{-}\widehat{B}_{+}^{3}B_{m}\widehat{E}_{4}\ \widehat{A}_{-}^{2}-15\widehat{B}_{-}^{4}A_{+}B_{m}\widehat{E}_{4}\widehat{A}_{-}^{2}-36B_{-}\ \widehat{B}_{-}\widehat{A}_{+}\widehat{B}_{+}^{2}A_{m}\widehat{E}_{6}\widehat{A}_{-}^{2}+36A_{-}\ \widehat{B}_{-}\widehat{A}_{+}\widehat{B}_{+}^{2}B_{m}\widehat{E}_{6}\widehat{A}_{-}^{2}-24B_{-}\ \widehat{B}_{-}\widehat{A}_{+}^{3}A_{m}\widehat{E}_{4}\widehat{E}_{6}\widehat{A}_{-}^{2}\\\ &+24A_{-}\ \widehat{B}_{-}\widehat{A}_{+}^{3}B_{m}\widehat{E}_{4}\widehat{E}_{6}\widehat{A}_{-}^{2}-18B_{-}\ \widehat{B}_{-}^{2}\widehat{A}_{+}^{3}A_{m}\widehat{E}_{4}^{2}\widehat{A}_{-}+18A_{-}\widehat{B}_{-}^{2}\ \widehat{A}_{+}^{3}B_{m}\widehat{E}_{4}^{2}\widehat{A}_{-}-18B_{-}\widehat{B}_{-}^{2}\widehat{A}_{+}\ \widehat{B}_{+}^{2}A_{m}\widehat{E}_{4}\widehat{A}_{-}+18A_{-}\widehat{B}_{-}^{2}\widehat{A}_{+}\ \widehat{B}_{+}^{2}B_{m}\widehat{E}_{4}\widehat{A}_{-}\\\ &-36B_{-}\widehat{B}_{-}^{2}\widehat{A}_{+}^{2}\ \widehat{B}_{+}A_{m}\widehat{E}_{6}\widehat{A}_{-}+36A_{-}\widehat{B}_{-}^{2}\widehat{A}_{+}^{2}\ \widehat{B}_{+}B_{m}\widehat{E}_{6}\widehat{A}_{-}+2B_{-}\widehat{B}_{-}^{3}\widehat{B}_{+}^{3}\ A_{m}-\widehat{B}_{-}^{6}B_{+}A_{m}-2A_{-}\widehat{B}_{-}^{3}\widehat{B}_{+}^{3}B_{m}+\widehat{B}_{-}^{6}\ A_{+}B_{m}-6B_{-}\widehat{B}_{-}^{3}\widehat{A}_{+}^{2}\widehat{B}_{+}A_{m}\widehat{E}_{4}+6A_{-}\ \widehat{B}_{-}^{3}\widehat{A}_{+}^{2}\widehat{B}_{+}B_{m}\widehat{E}_{4}-4B_{-}\widehat{B}_{-}^{3}\ \widehat{A}_{+}^{3}A_{m}\widehat{E}_{6}\\\ &+4A_{-}\widehat{B}_{-}^{3}\widehat{A}_{+}^{3}B_{m}\widehat{E}_{6}\Bigr{)}\end{array}$
[reverse arrows, labels*=1,2]G2 |  $\begin{array}[]{ll}&{1\over 2^{17}3^{14}}\Bigr{(}-9580220A_{-}^{3}\widehat{B}_{+}A_{m}^{2}E_{4}^{4}\widehat{A}_{m}A_{+}^{5}+10831372A_{-}\ B_{-}^{2}\widehat{B}_{+}A_{m}^{2}E_{4}^{3}\widehat{A}_{m}A_{+}^{5}+6561A_{-}^{3}\widehat{B}_{+}\ B_{m}^{2}E_{4}^{3}\widehat{A}_{m}A_{+}^{5}-19683A_{-}^{2}B_{-}\widehat{B}_{+}A_{m}B_{m}E_{4}^{3}\ \widehat{A}_{m}A_{+}^{5}+2187A_{-}B_{-}^{2}\widehat{B}_{+}B_{m}^{2}E_{4}^{2}\widehat{A}_{m}\ A_{+}^{5}+2187B_{-}^{3}\widehat{B}_{+}A_{m}B_{m}E_{4}^{2}\widehat{A}_{m}A_{+}^{5}-7776A_{-}^{3}\ \widehat{B}_{+}B_{m}^{2}E_{6}^{2}\widehat{A}_{m}A_{+}^{5}+23328A_{-}^{2}B_{-}\widehat{B}_{+}A_{m}\ B_{m}E_{6}^{2}\widehat{A}_{m}A_{+}^{5}+972B_{-}^{3}\widehat{B}_{+}B_{m}^{2}E_{6}\widehat{A}_{m}\ A_{+}^{5}-23443594A_{-}^{2}B_{-}\widehat{B}_{+}A_{m}^{2}E_{4}^{2}E_{6}\widehat{A}_{m}\ A_{+}^{5}\\\ &-2534430B_{-}^{3}\widehat{B}_{+}A_{m}^{2}E_{4}E_{6}\widehat{A}_{m}A_{+}^{5}+5832A_{-}\ B_{-}^{2}\widehat{B}_{+}A_{m}B_{m}E_{4}E_{6}\widehat{A}_{m}A_{+}^{5}+9580220A_{-}^{3}\ \widehat{A}_{+}A_{m}^{2}E_{4}^{4}\widehat{B}_{m}A_{+}^{5}-10831372A_{-}B_{-}^{2}\widehat{A}_{+}\ A_{m}^{2}E_{4}^{3}\widehat{B}_{m}A_{+}^{5}-4332693A_{-}^{3}\widehat{A}_{+}B_{m}^{2}E_{4}^{3}\ \widehat{B}_{m}A_{+}^{5}-20039805A_{-}^{2}B_{-}\widehat{A}_{+}A_{m}B_{m}E_{4}^{3}\widehat{B}_{m}\ A_{+}^{5}-2538503A_{-}B_{-}^{2}\widehat{A}_{+}B_{m}^{2}E_{4}^{2}\widehat{B}_{m}A_{+}^{5}-2153871\ B_{-}^{3}\widehat{A}_{+}A_{m}B_{m}E_{4}^{2}\widehat{B}_{m}A_{+}^{5}+7776A_{-}^{3}\widehat{A}_{+}\ B_{m}^{2}E_{6}^{2}\widehat{B}_{m}A_{+}^{5}-23328A_{-}^{2}B_{-}\widehat{A}_{+}A_{m}B_{m}E_{6}^{2}\ \widehat{B}_{m}A_{+}^{5}\\\ &+569114B_{-}^{3}\widehat{A}_{+}B_{m}^{2}E_{6}\widehat{B}_{m}\ A_{+}^{5}+23443594A_{-}^{2}B_{-}\widehat{A}_{+}A_{m}^{2}E_{4}^{2}E_{6}\widehat{B}_{m}\ A_{+}^{5}+14852808A_{-}^{3}\widehat{A}_{+}A_{m}B_{m}E_{4}^{2}E_{6}\widehat{B}_{m}\ A_{+}^{5}+2534430B_{-}^{3}\widehat{A}_{+}A_{m}^{2}E_{4}E_{6}\widehat{B}_{m}A_{+}^{5}+5425100\ A_{-}^{2}B_{-}\widehat{A}_{+}B_{m}^{2}E_{4}E_{6}\widehat{B}_{m}A_{+}^{5}+9740736A_{-}B_{-}^{2}\ \widehat{A}_{+}A_{m}B_{m}E_{4}E_{6}\widehat{B}_{m}A_{+}^{5}+4421216A_{-}^{2}B_{-}B_{+}\ \widehat{B}_{+}A_{m}^{2}E_{4}^{3}\widehat{A}_{m}A_{+}^{4}+6561A_{-}^{3}B_{+}\widehat{B}_{+}A_{m}\ B_{m}E_{4}^{3}\widehat{A}_{m}A_{+}^{4}+566820B_{-}^{3}B_{+}\widehat{B}_{+}A_{m}^{2}E_{4}^{2}\ \widehat{A}_{m}A_{+}^{4}\\\ &-4374A_{-}^{2}B_{-}B_{+}\widehat{B}_{+}B_{m}^{2}E_{4}^{2}\widehat{A}_{m}\ A_{+}^{4}+10935A_{-}B_{-}^{2}B_{+}\widehat{B}_{+}A_{m}B_{m}E_{4}^{2}\widehat{A}_{m}\ A_{+}^{4}-23328A_{-}^{2}B_{-}B_{+}\widehat{B}_{+}A_{m}^{2}E_{6}^{2}\widehat{A}_{m}A_{+}^{4}-7776\ A_{-}^{3}B_{+}\widehat{B}_{+}A_{m}B_{m}E_{6}^{2}\widehat{A}_{m}A_{+}^{4}+131220B_{-}^{3}B_{+}\ \widehat{B}_{+}B_{m}^{2}E_{4}\widehat{A}_{m}A_{+}^{4}+2916A_{-}B_{-}^{2}B_{+}\widehat{B}_{+}\ B_{m}^{2}E_{6}\widehat{A}_{m}A_{+}^{4}+49707166A_{-}^{3}B_{+}\widehat{B}_{+}A_{m}^{2}E_{4}^{2}\ E_{6}\widehat{A}_{m}A_{+}^{4}+3888B_{-}^{3}B_{+}\widehat{B}_{+}A_{m}B_{m}E_{6}\widehat{A}_{m}\ A_{+}^{4}+309202A_{-}B_{-}^{2}B_{+}\widehat{B}_{+}A_{m}^{2}E_{4}E_{6}\widehat{A}_{m}\ A_{+}^{4}\\\ &-5832A_{-}^{3}B_{+}\widehat{B}_{+}B_{m}^{2}E_{4}E_{6}\widehat{A}_{m}A_{+}^{4}+5832\ A_{-}^{2}B_{-}B_{+}\widehat{B}_{+}A_{m}B_{m}E_{4}E_{6}\widehat{A}_{m}A_{+}^{4}+15638272\ A_{-}^{2}B_{-}B_{+}\widehat{A}_{+}A_{m}^{2}E_{4}^{3}\widehat{B}_{m}A_{+}^{4}-6561A_{-}^{3}B_{+}\ \widehat{A}_{+}A_{m}B_{m}E_{4}^{3}\widehat{B}_{m}A_{+}^{4}+1584864B_{-}^{3}B_{+}\widehat{A}_{+}\ A_{m}^{2}E_{4}^{2}\widehat{B}_{m}A_{+}^{4}-1309112A_{-}^{2}B_{-}B_{+}\widehat{A}_{+}B_{m}^{2}\ E_{4}^{2}\widehat{B}_{m}A_{+}^{4}-10935A_{-}B_{-}^{2}B_{+}\widehat{A}_{+}A_{m}B_{m}E_{4}^{2}\ \widehat{B}_{m}A_{+}^{4}+23328A_{-}^{2}B_{-}B_{+}\widehat{A}_{+}A_{m}^{2}E_{6}^{2}\widehat{B}_{m}\ A_{+}^{4}+7776A_{-}^{3}B_{+}\widehat{A}_{+}A_{m}B_{m}E_{6}^{2}\widehat{B}_{m}A_{+}^{4}-149856\ B_{-}^{3}B_{+}\widehat{A}_{+}B_{m}^{2}E_{4}\widehat{B}_{m}A_{+}^{4}\\\ &+548360A_{-}B_{-}^{2}B_{+}\ \widehat{A}_{+}B_{m}^{2}E_{6}\widehat{B}_{m}A_{+}^{4}-64559974A_{-}^{3}B_{+}\widehat{A}_{+}\ A_{m}^{2}E_{4}^{2}E_{6}\widehat{B}_{m}A_{+}^{4}-3888B_{-}^{3}B_{+}\widehat{A}_{+}A_{m}B_{m}E_{6}\ \widehat{B}_{m}A_{+}^{4}-10055770A_{-}B_{-}^{2}B_{+}\widehat{A}_{+}A_{m}^{2}E_{4}E_{6}\ \widehat{B}_{m}A_{+}^{4}+1078988A_{-}^{3}B_{+}\widehat{A}_{+}B_{m}^{2}E_{4}E_{6}\widehat{B}_{m}\ A_{+}^{4}-5832A_{-}^{2}B_{-}B_{+}\widehat{A}_{+}A_{m}B_{m}E_{4}E_{6}\widehat{B}_{m}\ A_{+}^{4}-41022438A_{-}^{3}B_{+}^{2}\widehat{B}_{+}A_{m}^{2}E_{4}^{3}\widehat{A}_{m}\ A_{+}^{3}-1480438A_{-}B_{-}^{2}B_{+}^{2}\widehat{B}_{+}A_{m}^{2}E_{4}^{2}\widehat{A}_{m}\ A_{+}^{3}\\\ &+116640A_{-}^{3}B_{+}^{2}\widehat{B}_{+}B_{m}^{2}E_{4}^{2}\widehat{A}_{m}A_{+}^{3}+4374\ A_{-}^{2}B_{-}B_{+}^{2}\widehat{B}_{+}A_{m}B_{m}E_{4}^{2}\widehat{A}_{m}A_{+}^{3}+15552A_{-}^{3}\ B_{+}^{2}\widehat{B}_{+}A_{m}^{2}E_{6}^{2}\widehat{A}_{m}A_{+}^{3}-2916A_{-}B_{-}^{2}B_{+}^{2}\ \widehat{B}_{+}B_{m}^{2}E_{4}\widehat{A}_{m}A_{+}^{3}+1458B_{-}^{3}B_{+}^{2}\widehat{B}_{+}A_{m}\ B_{m}E_{4}\widehat{A}_{m}A_{+}^{3}-107198B_{-}^{3}B_{+}^{2}\widehat{B}_{+}A_{m}^{2}E_{6}\ \widehat{A}_{m}A_{+}^{3}+43740A_{-}^{2}B_{-}B_{+}^{2}\widehat{B}_{+}B_{m}^{2}E_{6}\widehat{A}_{m}\ A_{+}^{3}+5832A_{-}B_{-}^{2}B_{+}^{2}\widehat{B}_{+}A_{m}B_{m}E_{6}\widehat{A}_{m}\ A_{+}^{3}-2151656A_{-}^{2}B_{-}B_{+}^{2}\widehat{B}_{+}A_{m}^{2}E_{4}E_{6}\widehat{A}_{m}\ A_{+}^{3}\\\ &+45348570A_{-}^{3}B_{+}^{2}\widehat{A}_{+}A_{m}^{2}E_{4}^{3}\widehat{B}_{m}\ A_{+}^{3}+4016754A_{-}B_{-}^{2}B_{+}^{2}\widehat{A}_{+}A_{m}^{2}E_{4}^{2}\widehat{B}_{m}\ A_{+}^{3}+187996A_{-}^{3}B_{+}^{2}\widehat{A}_{+}B_{m}^{2}E_{4}^{2}\widehat{B}_{m}A_{+}^{3}-4374\ A_{-}^{2}B_{-}B_{+}^{2}\widehat{A}_{+}A_{m}B_{m}E_{4}^{2}\widehat{B}_{m}A_{+}^{3}-15552A_{-}^{3}\ B_{+}^{2}\widehat{A}_{+}A_{m}^{2}E_{6}^{2}\widehat{B}_{m}A_{+}^{3}-48638A_{-}B_{-}^{2}B_{+}^{2}\ \widehat{A}_{+}B_{m}^{2}E_{4}\widehat{B}_{m}A_{+}^{3}-1458B_{-}^{3}B_{+}^{2}\widehat{A}_{+}A_{m}\ B_{m}E_{4}\widehat{B}_{m}A_{+}^{3}-462888B_{-}^{3}B_{+}^{2}\widehat{A}_{+}A_{m}^{2}E_{6}\ \widehat{B}_{m}A_{+}^{3}+127532A_{-}^{2}B_{-}B_{+}^{2}\widehat{A}_{+}B_{m}^{2}E_{6}\widehat{B}_{m}\ A_{+}^{3}\\\ &-5832A_{-}B_{-}^{2}B_{+}^{2}\widehat{A}_{+}A_{m}B_{m}E_{6}\widehat{B}_{m}\ A_{+}^{3}-3273444A_{-}^{2}B_{-}B_{+}^{2}\widehat{A}_{+}A_{m}^{2}E_{4}E_{6}\widehat{B}_{m}\ A_{+}^{3}-486B_{-}^{3}B_{+}^{3}\widehat{B}_{+}B_{m}^{2}\widehat{A}_{m}A_{+}^{2}+1269946A_{-}^{2}\ B_{-}B_{+}^{3}\widehat{B}_{+}A_{m}^{2}E_{4}^{2}\widehat{A}_{m}A_{+}^{2}+4374A_{-}^{3}B_{+}^{3}\ \widehat{B}_{+}A_{m}B_{m}E_{4}^{2}\widehat{A}_{m}A_{+}^{2}+86160B_{-}^{3}B_{+}^{3}\widehat{B}_{+}\ A_{m}^{2}E_{4}\widehat{A}_{m}A_{+}^{2}-4374A_{-}^{2}B_{-}B_{+}^{3}\widehat{B}_{+}B_{m}^{2}E_{4}\ \widehat{A}_{m}A_{+}^{2}+1458A_{-}B_{-}^{2}B_{+}^{3}\widehat{B}_{+}A_{m}B_{m}E_{4}\widehat{A}_{m}\ A_{+}^{2}+364976A_{-}B_{-}^{2}B_{+}^{3}\widehat{B}_{+}A_{m}^{2}E_{6}\widehat{A}_{m}A_{+}^{2}-4860\ A_{-}^{3}B_{+}^{3}\widehat{B}_{+}B_{m}^{2}E_{6}\widehat{A}_{m}A_{+}^{2}\\\ &+5832A_{-}^{2}B_{-}B_{+}^{3}\ \widehat{B}_{+}A_{m}B_{m}E_{6}\widehat{A}_{m}A_{+}^{2}+16676780A_{-}^{3}B_{+}^{3}\widehat{B}_{+}\ A_{m}^{2}E_{4}E_{6}\widehat{A}_{m}A_{+}^{2}+338B_{-}^{3}B_{+}^{3}\widehat{A}_{+}B_{m}^{2}\ \widehat{B}_{m}A_{+}^{2}+43540A_{-}^{2}B_{-}B_{+}^{3}\widehat{A}_{+}A_{m}^{2}E_{4}^{2}\widehat{B}_{m}\ A_{+}^{2}-4374A_{-}^{3}B_{+}^{3}\widehat{A}_{+}A_{m}B_{m}E_{4}^{2}\widehat{B}_{m}A_{+}^{2}-67524\ B_{-}^{3}B_{+}^{3}\widehat{A}_{+}A_{m}^{2}E_{4}\widehat{B}_{m}A_{+}^{2}+4374A_{-}^{2}B_{-}B_{+}^{3}\ \widehat{A}_{+}B_{m}^{2}E_{4}\widehat{B}_{m}A_{+}^{2}-1458A_{-}B_{-}^{2}B_{+}^{3}\widehat{A}_{+}\ A_{m}B_{m}E_{4}\widehat{B}_{m}A_{+}^{2}-916252A_{-}B_{-}^{2}B_{+}^{3}\widehat{A}_{+}A_{m}^{2}\ E_{6}\widehat{B}_{m}A_{+}^{2}\\\ &-266062A_{-}^{3}B_{+}^{3}\widehat{A}_{+}B_{m}^{2}E_{6}\widehat{B}_{m}\ A_{+}^{2}-5832A_{-}^{2}B_{-}B_{+}^{3}\widehat{A}_{+}A_{m}B_{m}E_{6}\widehat{B}_{m}\ A_{+}^{2}-17749936A_{-}^{3}B_{+}^{3}\widehat{A}_{+}A_{m}^{2}E_{4}E_{6}\widehat{B}_{m}\ A_{+}^{2}+729A_{-}B_{-}^{2}B_{+}^{4}\widehat{B}_{+}B_{m}^{2}\widehat{A}_{m}A_{+}-3698028A_{-}^{3}\ B_{+}^{4}\widehat{B}_{+}A_{m}^{2}E_{4}^{2}\widehat{A}_{m}A_{+}+243B_{-}^{3}B_{+}^{4}\widehat{B}_{+}\ A_{m}B_{m}\widehat{A}_{m}A_{+}-133668A_{-}B_{-}^{2}B_{+}^{4}\widehat{B}_{+}A_{m}^{2}E_{4}\ \widehat{A}_{m}A_{+}-729A_{-}^{3}B_{+}^{4}\widehat{B}_{+}B_{m}^{2}E_{4}\widehat{A}_{m}A_{+}+3645\ A_{-}^{2}B_{-}B_{+}^{4}\widehat{B}_{+}A_{m}B_{m}E_{4}\widehat{A}_{m}A_{+}+1116A_{-}^{2}B_{-}\ B_{+}^{4}\widehat{B}_{+}A_{m}^{2}E_{6}\widehat{A}_{m}A_{+}\\\ &+3888A_{-}^{3}B_{+}^{4}\widehat{B}_{+}\ A_{m}B_{m}E_{6}\widehat{A}_{m}A_{+}+3393392A_{-}^{3}B_{+}^{4}\widehat{A}_{+}A_{m}^{2}E_{4}^{2}\ \widehat{B}_{m}A_{+}-243B_{-}^{3}B_{+}^{4}\widehat{A}_{+}A_{m}B_{m}\widehat{B}_{m}A_{+}+185222\ A_{-}B_{-}^{2}B_{+}^{4}\widehat{A}_{+}A_{m}^{2}E_{4}\widehat{B}_{m}A_{+}+93312A_{-}^{3}B_{+}^{4}\ \widehat{A}_{+}B_{m}^{2}E_{4}\widehat{B}_{m}A_{+}-3645A_{-}^{2}B_{-}B_{+}^{4}\widehat{A}_{+}A_{m}\ B_{m}E_{4}\widehat{B}_{m}A_{+}-172388A_{-}^{2}B_{-}B_{+}^{4}\widehat{A}_{+}A_{m}^{2}E_{6}\ \widehat{B}_{m}A_{+}-3888A_{-}^{3}B_{+}^{4}\widehat{A}_{+}A_{m}B_{m}E_{6}\widehat{B}_{m}\ A_{+}+391B_{-}^{3}B_{+}^{5}\widehat{B}_{+}A_{m}^{2}\widehat{A}_{m}-729A_{-}B_{-}^{2}B_{+}^{5}\ \widehat{B}_{+}A_{m}B_{m}\widehat{A}_{m}\\\ &+729A_{-}^{2}B_{-}B_{+}^{5}\widehat{B}_{+}A_{m}^{2}E_{4}\ \widehat{A}_{m}+729A_{-}^{3}B_{+}^{5}\widehat{B}_{+}A_{m}B_{m}E_{4}\widehat{A}_{m}+271894\ A_{-}^{3}B_{+}^{5}\widehat{B}_{+}A_{m}^{2}E_{6}\widehat{A}_{m}-243B_{-}^{3}B_{+}^{5}\widehat{A}_{+}\ A_{m}^{2}\widehat{B}_{m}-729A_{-}^{2}B_{-}B_{+}^{5}\widehat{A}_{+}A_{m}^{2}E_{4}\ \widehat{B}_{m}-93312A_{-}^{3}B_{+}^{5}\widehat{A}_{+}A_{m}B_{m}E_{4}\widehat{B}_{m}-972A_{-}^{3}\ B_{+}^{5}\widehat{A}_{+}A_{m}^{2}E_{6}\widehat{B}_{m}\Bigr{)}\end{array}$
Table 9: Elliptic genera of $G_{2}$. Here, $A_{z}=A_{z}(\tau)$, $\widehat{A}_{z}=A_{z}(3\tau)$, and same for $B_{z}$. $\mathfrak{n}$ | $\mathcal{N}$
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[labels*=1,1,1]B3 | ${1\over 2^{6}3^{3}}(A_{m}B_{-}-A_{-}B_{m})^{2}(\widehat{A}_{+}\widehat{B}_{m}-\widehat{A}_{m}\widehat{A}_{+})$
[labels*=2,1,1]B3 |  $\begin{array}[]{ll}&{1\over 2^{19}3^{10}}(A_{m}B_{+}-A_{+}B_{m})(A_{m}B_{-}-A_{-}B_{m})(\widehat{A}_{m}\widehat{B}_{+}-\widehat{A}_{+}\widehat{B}_{m})\times\Bigr{(}81A_{-}^{2}A_{+}^{4}B_{-}E_{4}^{3}A_{m}-27A_{-}^{3}A_{+}^{4}E_{4}^{3}B_{m}-9A_{+}^{4}B_{-}^{3}E_{4}^{2}A_{m}-9A_{-}A_{+}^{4}B_{-}^{2}E_{4}^{2}B_{m}-96A_{-}^{2}A_{+}^{4}B_{-}E_{6}^{2}A_{m}+32A_{-}^{3}A_{+}^{4}E_{6}^{2}B_{m}+4A_{+}^{4}B_{-}^{3}E_{6}B_{m}\\\ &+24A_{-}A_{+}^{4}B_{-}^{2}E_{4}E_{6}A_{m}-54A_{-}^{3}A_{+}^{3}B_{+}E_{4}^{3}A_{m}-54A_{-}A_{+}^{3}B_{-}^{2}B_{+}E_{4}^{2}A_{m}+18A_{-}^{2}A_{+}^{3}B_{-}B_{+}E_{4}^{2}B_{m}+64A_{-}^{3}A_{+}^{3}B_{+}E_{6}^{2}A_{m}-6A_{+}^{3}B_{-}^{3}B_{+}E_{4}B_{m}+20A_{+}^{3}B_{-}^{3}B_{+}E_{6}A_{m}+12A_{-}A_{+}^{3}B_{-}^{2}B_{+}E_{6}B_{m}+24A_{-}^{2}A_{+}^{3}B_{-}B_{+}E_{4}E_{6}A_{m}-24A_{-}^{3}A_{+}^{3}B_{+}E_{4}E_{6}B_{m}\\\ &+36A_{-}^{3}A_{+}^{2}B_{+}^{2}E_{4}^{2}B_{m}-12A_{+}^{2}B_{-}^{3}B_{+}^{2}E_{4}A_{m}+36A_{-}A_{+}^{2}B_{-}^{2}B_{+}^{2}E_{6}A_{m}-36A_{-}^{2}A_{+}^{2}B_{-}B_{+}^{2}E_{6}B_{m}-24A_{-}^{3}A_{+}^{2}B_{+}^{2}E_{4}E_{6}A_{m}+18A_{-}^{3}A_{+}B_{+}^{3}E_{4}^{2}A_{m}-6A_{-}A_{+}B_{-}^{2}B_{+}^{3}E_{4}A_{m}+18A_{-}^{2}A_{+}B_{-}B_{+}^{3}E_{4}B_{m}-12A_{-}^{2}A_{+}B_{-}B_{+}^{3}E_{6}A_{m}\\\ &-20A_{-}^{3}A_{+}B_{+}^{3}E_{6}B_{m}+3A_{-}^{2}B_{-}B_{+}^{4}E_{4}A_{m}+3A_{-}^{3}B_{+}^{4}E_{4}B_{m}-4A_{-}^{3}B_{+}^{4}E_{6}A_{m}+2A_{+}B_{-}^{3}B_{+}^{3}B_{m}+B_{-}^{3}B_{+}^{4}A_{m}-3A_{-}B_{-}^{2}B_{+}^{4}B_{m}\Bigr{)}\end{array}$
[labels*=1,2,1]B3 |  $\begin{array}[]{ll}&{1\over 2^{19}3^{10}}(A_{m}B_{+}-A_{+}B_{m})(A_{m}B_{-}-A_{-}B_{m})(\widehat{A}_{m}\widehat{B}_{+}-\widehat{A}_{+}\widehat{B}_{m})\times\Bigr{(}-81A_{+}^{2}A_{-}^{4}B_{+}E_{4}^{3}A_{m}-27A_{+}^{3}A_{-}^{4}E_{4}^{3}B_{m}+9A_{-}^{4}B_{+}^{3}E_{4}^{2}A_{m}+27A_{+}A_{-}^{4}B_{+}^{2}E_{4}^{2}B_{m}+96A_{+}^{2}A_{-}^{4}B_{+}E_{6}^{2}A_{m}+32A_{+}^{3}A_{-}^{4}E_{6}^{2}B_{m}-8A_{-}^{4}B_{+}^{3}E_{6}B_{m}\\\ &-24A_{+}A_{-}^{4}B_{+}^{2}E_{4}E_{6}A_{m}-24A_{+}^{2}A_{-}^{4}B_{+}E_{4}E_{6}B_{m}+108A_{+}^{3}A_{-}^{3}B_{-}E_{4}^{3}A_{m}+36A_{+}A_{-}^{3}B_{-}B_{+}^{2}E_{4}^{2}A_{m}+36A_{+}^{2}A_{-}^{3}B_{-}B_{+}E_{4}^{2}B_{m}-128A_{+}^{3}A_{-}^{3}B_{-}E_{6}^{2}A_{m}+12A_{-}^{3}B_{-}B_{+}^{3}E_{4}B_{m}-16A_{-}^{3}B_{-}B_{+}^{3}E_{6}A_{m}-48A_{+}A_{-}^{3}B_{-}B_{+}^{2}E_{6}B_{m}\\\ &-54A_{+}^{2}A_{-}^{2}B_{-}^{2}B_{+}E_{4}^{2}A_{m}-18A_{+}^{3}A_{-}^{2}B_{-}^{2}E_{4}^{2}B_{m}+6A_{-}^{2}B_{-}^{2}B_{+}^{3}E_{4}A_{m}+18A_{+}A_{-}^{2}B_{-}^{2}B_{+}^{2}E_{4}B_{m}+48A_{+}^{3}A_{-}^{2}B_{-}^{2}E_{4}E_{6}A_{m}-36A_{+}^{3}A_{-}B_{-}^{3}E_{4}^{2}A_{m}-12A_{+}A_{-}B_{-}^{3}B_{+}^{2}E_{4}A_{m}\\\ &-12A_{+}^{2}A_{-}B_{-}^{3}B_{+}E_{4}B_{m}+48A_{+}^{2}A_{-}B_{-}^{3}B_{+}E_{6}A_{m}+16A_{+}^{3}A_{-}B_{-}^{3}E_{6}B_{m}-9A_{+}^{2}B_{-}^{4}B_{+}E_{4}A_{m}-3A_{+}^{3}B_{-}^{4}E_{4}B_{m}+8A_{+}^{3}B_{-}^{4}E_{6}A_{m}-4A_{-}B_{-}^{3}B_{+}^{3}B_{m}+B_{-}^{4}B_{+}^{3}A_{m}+3A_{+}B_{-}^{4}B_{+}^{2}B_{m}\Bigr{)}\end{array}$
Table 10: Elliptic genera of $B_{3}$. Here, $A_{z}=A_{z}(\tau)$, $\widehat{A}_{z}=A_{z}(2\tau)$, and same for $B_{z}$. $\mathfrak{n}$ | $\mathcal{N}$
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[labels*=1,1,1]C3 | ${1\over 2^{6}3^{3}}(A_{-}B_{m}-A_{m}B_{-})(\widehat{A}_{-}\widehat{B}_{m}-\widehat{A}_{m}\widehat{B}_{-})(\widehat{A}_{+}\widehat{B}_{m}-\widehat{A}_{m}\widehat{B}_{+})$
[labels*=1,1,2]C3 |  $\begin{array}[]{ll}&{1\over 2^{19}3^{10}}(A_{m}B_{+}-A_{+}B_{m})(\widehat{A}_{m}\widehat{B}_{+}-\widehat{A}_{+}\widehat{B}_{m})(\widehat{A}_{m}\widehat{B}_{-}-\widehat{A}_{-}\widehat{B}_{m})\Bigr{(}81A_{-}^{2}A_{+}^{4}B_{-}E_{4}^{3}A_{m}-27A_{-}^{3}A_{+}^{4}E_{4}^{3}B_{m}-9A_{+}^{4}B_{-}^{3}E_{4}^{2}A_{m}-9A_{-}A_{+}^{4}B_{-}^{2}E_{4}^{2}B_{m}-96A_{-}^{2}A_{+}^{4}B_{-}E_{6}^{2}A_{m}+32A_{-}^{3}A_{+}^{4}E_{6}^{2}B_{m}+4A_{+}^{4}B_{-}^{3}E_{6}B_{m}+24A_{-}A_{+}^{4}B_{-}^{2}E_{4}E_{6}A_{m}\\\ &-54A_{-}^{3}A_{+}^{3}B_{+}E_{4}^{3}A_{m}-54A_{-}A_{+}^{3}B_{-}^{2}B_{+}E_{4}^{2}A_{m}+18A_{-}^{2}A_{+}^{3}B_{-}B_{+}E_{4}^{2}B_{m}+64A_{-}^{3}A_{+}^{3}B_{+}E_{6}^{2}A_{m}-6A_{+}^{3}B_{-}^{3}B_{+}E_{4}B_{m}+20A_{+}^{3}B_{-}^{3}B_{+}E_{6}A_{m}+12A_{-}A_{+}^{3}B_{-}^{2}B_{+}E_{6}B_{m}+24A_{-}^{2}A_{+}^{3}B_{-}B_{+}E_{4}E_{6}A_{m}-24A_{-}^{3}A_{+}^{3}B_{+}E_{4}E_{6}B_{m}\\\ &+36A_{-}^{3}A_{+}^{2}B_{+}^{2}E_{4}^{2}B_{m}-12A_{+}^{2}B_{-}^{3}B_{+}^{2}E_{4}A_{m}+36A_{-}A_{+}^{2}B_{-}^{2}B_{+}^{2}E_{6}A_{m}-36A_{-}^{2}A_{+}^{2}B_{-}B_{+}^{2}E_{6}B_{m}-24A_{-}^{3}A_{+}^{2}B_{+}^{2}E_{4}E_{6}A_{m}+18A_{-}^{3}A_{+}B_{+}^{3}E_{4}^{2}A_{m}-6A_{-}A_{+}B_{-}^{2}B_{+}^{3}E_{4}A_{m}+18A_{-}^{2}A_{+}B_{-}B_{+}^{3}E_{4}B_{m}-12A_{-}^{2}A_{+}B_{-}B_{+}^{3}E_{6}A_{m}\\\ &-20A_{-}^{3}A_{+}B_{+}^{3}E_{6}B_{m}+3A_{-}^{2}B_{-}B_{+}^{4}E_{4}A_{m}+3A_{-}^{3}B_{+}^{4}E_{4}B_{m}-4A_{-}^{3}B_{+}^{4}E_{6}A_{m}+2A_{+}B_{-}^{3}B_{+}^{3}B_{m}+B_{-}^{3}B_{+}^{4}A_{m}-3A_{-}B_{-}^{2}B_{+}^{4}B_{m}\Bigr{)}\end{array}$
Table 11: Elliptic genera of $(C_{3})_{0}$. Here, $A_{z}=A_{z}(\tau)$, $\widehat{A}_{z}=A_{z}(2\tau)$, and same for $B_{z}$. $\mathfrak{n}$ | $\mathcal{N}$
---|---
[labels*=1,1,1,1]F4 | ${1\over 2^{8}3^{4}}(\widehat{A}_{m}\widehat{B}_{+}-\widehat{A}_{+}\widehat{B}_{m})(\widehat{A}_{m}\widehat{B}_{-}-\widehat{A}_{-}\widehat{B}_{m})(A_{m}B_{-}-A_{-}B_{m})^{2}$
[labels*=2,1,1,1]F4 |  $\begin{array}[]{ll}&{1\over 2^{21}3^{11}}(A_{m}B_{-}-A_{-}B_{m})(A_{m}B_{+}-A_{+}B_{m})(\widehat{A}_{m}\widehat{B}_{-}-\widehat{A}_{-}\widehat{B}_{m})(\widehat{A}_{m}\widehat{B}_{+}-\widehat{A}_{+}\widehat{B}_{m})\times\Bigr{(}81A_{-}^{2}A_{+}^{4}B_{-}E_{4}^{3}A_{m}-27A_{-}^{3}A_{+}^{4}E_{4}^{3}B_{m}-9A_{+}^{4}B_{-}^{3}E_{4}^{2}A_{m}-9A_{-}A_{+}^{4}B_{-}^{2}E_{4}^{2}B_{m}\\\ &-96A_{-}^{2}A_{+}^{4}B_{-}E_{6}^{2}A_{m}+32A_{-}^{3}A_{+}^{4}E_{6}^{2}B_{m}-4A_{+}^{4}B_{-}^{3}E_{6}B_{m}-24A_{-}A_{+}^{4}B_{-}^{2}E_{4}E_{6}A_{m}-54A_{-}^{3}A_{+}^{3}B_{+}E_{4}^{3}A_{m}-54A_{-}A_{+}^{3}B_{-}^{2}B_{+}E_{4}^{2}A_{m}+18A_{-}^{2}A_{+}^{3}B_{-}B_{+}E_{4}^{2}B_{m}+64A_{-}^{3}A_{+}^{3}B_{+}E_{6}^{2}A_{m}\\\ &-6A_{+}^{3}B_{-}^{3}B_{+}E_{4}B_{m}-20A_{+}^{3}B_{-}^{3}B_{+}E_{6}A_{m}-12A_{-}A_{+}^{3}B_{-}^{2}B_{+}E_{6}B_{m}-24A_{-}^{2}A_{+}^{3}B_{-}B_{+}E_{4}E_{6}A_{m}+24A_{-}^{3}A_{+}^{3}B_{+}E_{4}E_{6}B_{m}+36A_{-}^{3}A_{+}^{2}B_{+}^{2}E_{4}^{2}B_{m}-12A_{+}^{2}B_{-}^{3}B_{+}^{2}E_{4}A_{m}-36A_{-}A_{+}^{2}B_{-}^{2}B_{+}^{2}E_{6}A_{m}+36A_{-}^{2}A_{+}^{2}B_{-}B_{+}^{2}E_{6}B_{m}\\\ &+24A_{-}^{3}A_{+}^{2}B_{+}^{2}E_{4}E_{6}A_{m}+18A_{-}^{3}A_{+}B_{+}^{3}E_{4}^{2}A_{m}-6A_{-}A_{+}B_{-}^{2}B_{+}^{3}E_{4}A_{m}+18A_{-}^{2}A_{+}B_{-}B_{+}^{3}E_{4}B_{m}+12A_{-}^{2}A_{+}B_{-}B_{+}^{3}E_{6}A_{m}+20A_{-}^{3}A_{+}B_{+}^{3}E_{6}B_{m}+3A_{-}^{2}B_{-}B_{+}^{4}E_{4}A_{m}+3A_{-}^{3}B_{+}^{4}E_{4}B_{m}\\\ &+4A_{-}^{3}B_{+}^{4}E_{6}A_{m}+2A_{+}B_{-}^{3}B_{+}^{3}B_{m}+B_{-}^{3}B_{+}^{4}A_{m}-3A_{-}B_{-}^{2}B_{+}^{4}B_{m}\Bigr{)}\end{array}$
[labels*=1,2,1,1]F4 |  $\begin{array}[]{ll}&{1\over 2^{21}3^{11}}(A_{m}B_{-}-A_{-}B_{m})(A_{m}B_{+}-A_{+}B_{m})(\widehat{A}_{m}\widehat{B}_{-}-\widehat{A}_{-}\widehat{B}_{m})(\widehat{A}_{m}\widehat{B}_{+}-\widehat{A}_{+}\widehat{B}_{m})\times\Bigr{(}-81A_{+}^{2}A_{-}^{4}B_{+}E_{4}^{3}A_{m}-27A_{+}^{3}A_{-}^{4}E_{4}^{3}B_{m}+9A_{-}^{4}B_{+}^{3}E_{4}^{2}A_{m}+27A_{+}A_{-}^{4}B_{+}^{2}E_{4}^{2}B_{m}\\\ &+96A_{+}^{2}A_{-}^{4}B_{+}E_{6}^{2}A_{m}+32A_{+}^{3}A_{-}^{4}E_{6}^{2}B_{m}+8A_{-}^{4}B_{+}^{3}E_{6}B_{m}+24A_{+}A_{-}^{4}B_{+}^{2}E_{4}E_{6}A_{m}+24A_{+}^{2}A_{-}^{4}B_{+}E_{4}E_{6}B_{m}+108A_{+}^{3}A_{-}^{3}B_{-}E_{4}^{3}A_{m}+36A_{+}A_{-}^{3}B_{-}B_{+}^{2}E_{4}^{2}A_{m}+36A_{+}^{2}A_{-}^{3}B_{-}B_{+}E_{4}^{2}B_{m}-128A_{+}^{3}A_{-}^{3}B_{-}E_{6}^{2}A_{m}\\\ &+12A_{-}^{3}B_{-}B_{+}^{3}E_{4}B_{m}+16A_{-}^{3}B_{-}B_{+}^{3}E_{6}A_{m}+48A_{+}A_{-}^{3}B_{-}B_{+}^{2}E_{6}B_{m}-54A_{+}^{2}A_{-}^{2}B_{-}^{2}B_{+}E_{4}^{2}A_{m}-18A_{+}^{3}A_{-}^{2}B_{-}^{2}E_{4}^{2}B_{m}+6A_{-}^{2}B_{-}^{2}B_{+}^{3}E_{4}A_{m}+18A_{+}A_{-}^{2}B_{-}^{2}B_{+}^{2}E_{4}B_{m}-48A_{+}^{3}A_{-}^{2}B_{-}^{2}E_{4}E_{6}A_{m}-36A_{+}^{3}A_{-}B_{-}^{3}E_{4}^{2}A_{m}\\\ &-12A_{+}A_{-}B_{-}^{3}B_{+}^{2}E_{4}A_{m}-12A_{+}^{2}A_{-}B_{-}^{3}B_{+}E_{4}B_{m}-48A_{+}^{2}A_{-}B_{-}^{3}B_{+}E_{6}A_{m}-16A_{+}^{3}A_{-}B_{-}^{3}E_{6}B_{m}-9A_{+}^{2}B_{-}^{4}B_{+}E_{4}A_{m}-3A_{+}^{3}B_{-}^{4}E_{4}B_{m}-8A_{+}^{3}B_{-}^{4}E_{6}A_{m}-4A_{-}B_{-}^{3}B_{+}^{3}B_{m}+B_{-}^{4}B_{+}^{3}A_{m}+3A_{+}B_{-}^{4}B_{+}^{2}B_{m}\Bigr{)}\end{array}$
Table 12: Elliptic genera of $F_{4}$. Here, $A_{z}=A_{z}(\tau)$, $\widehat{A}_{z}=A_{z}(2\tau)$, and same for $B_{z}$. $\mathfrak{n}$ | $\mathcal{N}$
---|---
[labels*=1]A1 | $\frac{1}{2^{8}3^{2}}(\widehat{A}_{m}\widehat{B}_{+}-\widehat{A}_{+}\widehat{B}_{m})(\widehat{A}_{m}\widehat{B}_{-}-\widehat{A}_{-}\widehat{B}_{m})(E_{2}^{(2)}-E_{2}^{(4)})$
[labels*=1,0]C2 | ${1\over 12}(A_{m}B_{+}-A_{+}B_{m})$
[labels*=0,1]C2 | ${\frac{1}{2^{8}3^{2}}}(\widehat{A}_{m}\widehat{B}_{+}-\widehat{A}_{+}\widehat{B}_{m})(\widehat{A}_{m}\widehat{B}_{-}-\widehat{A}_{-}\widehat{B}_{m})(E_{2}^{(2)}-E_{2}^{(4)})$
[labels*=1,1]C2 | ${\frac{1}{2^{10}3^{3}}}(A_{m}B_{-}-A_{-}B_{m})(\widehat{A}_{m}\widehat{B}_{+}-\widehat{A}_{+}\widehat{B}_{m})(\widehat{A}_{m}\widehat{B}_{-}-\widehat{A}_{-}\widehat{B}_{m})(E_{2}^{(2)}-E_{2}^{(4)})$
|
$\mathfrak{n}$ | $\mathcal{I}(\tau,\epsilon_{1},\epsilon_{2})$
[labels*=2]A1 |  $\begin{array}[]{ll}&{1\over 2^{63}3^{16}}\Bigr{(}E_{2,2}-E_{2,4}\Bigr{)}^{2}\Bigr{(}8E_{2,2}^{3}\widehat{A}_{-}\widehat{A}_{+}^{2}\widehat{B}_{-}+8E_{2,2}^{3}\widehat{A}_{-}^{2}\widehat{A}_{+}\widehat{B}_{+}+16E_{2,2}^{2}\widehat{A}_{+}^{2}\widehat{B}_{-}^{2}+16E_{2,2}^{2}\widehat{A}_{-}^{2}\widehat{B}_{+}^{2}+64E_{2,2}^{2}\widehat{A}_{-}\widehat{A}_{+}\widehat{B}_{-}\widehat{B}_{+}+128E_{2,2}\widehat{A}_{-}\widehat{B}_{-}\widehat{B}_{+}^{2}+128E_{2,2}\widehat{A}_{+}\widehat{B}_{-}^{2}\widehat{B}_{+}+E_{2,2}^{4}\widehat{A}_{-}^{2}\widehat{A}_{+}^{2}+256\widehat{B}_{-}^{2}\widehat{B}_{+}^{2}\\\ &+1224E_{2,4}E_{2,2}^{2}\widehat{A}_{-}\widehat{A}_{+}^{2}\widehat{B}_{-}+1224E_{2,4}E_{2,2}^{2}\widehat{A}_{-}^{2}\widehat{A}_{+}\widehat{B}_{+}+96E_{2,4}E_{2,2}\widehat{A}_{+}^{2}\widehat{B}_{-}^{2}+96E_{2,4}E_{2,2}\widehat{A}_{-}^{2}\widehat{B}_{+}^{2}-1920E_{2,4}E_{2,2}\widehat{A}_{-}\widehat{A}_{+}\widehat{B}_{-}\widehat{B}_{+}+144E_{2,4}^{2}\widehat{A}_{+}^{2}\widehat{B}_{-}^{2}+384E_{2,4}\widehat{A}_{-}\widehat{B}_{-}\widehat{B}_{+}^{2}+384E_{2,4}\widehat{A}_{+}\widehat{B}_{-}^{2}\widehat{B}_{+}-564E_{2,4}E_{2,2}^{3}\widehat{A}_{-}^{2}\widehat{A}_{+}^{2}\\\ &+3672E_{2,4}^{3}\widehat{A}_{-}\widehat{A}_{+}^{2}\widehat{B}_{-}+3672E_{2,4}^{3}\widehat{A}_{-}^{2}\widehat{A}_{+}\widehat{B}_{+}+144E_{2,4}^{2}\widehat{A}_{-}^{2}\widehat{B}_{+}^{2}+2880E_{2,4}^{2}\widehat{A}_{-}\widehat{A}_{+}\widehat{B}_{-}\widehat{B}_{+}-4392E_{2,2}E_{2,4}^{2}\widehat{A}_{-}\widehat{A}_{+}^{2}\widehat{B}_{-}-4392E_{2,2}E_{2,4}^{2}\widehat{A}_{-}^{2}\widehat{A}_{+}\widehat{B}_{+}+5265E_{2,4}^{4}\widehat{A}_{-}^{2}\widehat{A}_{+}^{2}-8532E_{2,2}E_{2,4}^{3}\widehat{A}_{-}^{2}\widehat{A}_{+}^{2}+4086E_{2,2}^{2}E_{2,4}^{2}\widehat{A}_{-}^{2}\widehat{A}_{+}^{2}\Bigr{)}\\\ &\Bigr{(}152E_{2,2}^{3}\widehat{A}_{-}\widehat{A}_{+}^{2}\widehat{B}_{-}+152E_{2,2}^{3}\widehat{A}_{-}^{2}\widehat{A}_{+}\widehat{B}_{+}+400E_{2,2}^{2}\widehat{A}_{+}^{2}\widehat{B}_{-}^{2}+400E_{2,2}^{2}\widehat{A}_{-}^{2}\widehat{B}_{+}^{2}-704E_{2,2}^{2}\widehat{A}_{-}\widehat{A}_{+}\widehat{B}_{-}\widehat{B}_{+}-640E_{2,2}\widehat{A}_{-}\widehat{B}_{-}\widehat{B}_{+}^{2}-640E_{2,2}\widehat{A}_{+}\widehat{B}_{-}^{2}\widehat{B}_{+}+49E_{2,2}^{4}\widehat{A}_{-}^{2}\widehat{A}_{+}^{2}+256\widehat{B}_{-}^{2}\widehat{B}_{+}^{2}-1512E_{2,4}E_{2,2}^{2}\widehat{A}_{-}\widehat{A}_{+}^{2}\widehat{B}_{-}\\\ &-1512E_{2,4}E_{2,2}^{2}\widehat{A}_{-}^{2}\widehat{A}_{+}\widehat{B}_{+}-1440E_{2,4}E_{2,2}\widehat{A}_{+}^{2}\widehat{B}_{-}^{2}-1440E_{2,4}E_{2,2}\widehat{A}_{-}^{2}\widehat{B}_{+}^{2}+1152E_{2,4}E_{2,2}\widehat{A}_{-}\widehat{A}_{+}\widehat{B}_{-}\widehat{B}_{+}+1296E_{2,4}^{2}\widehat{A}_{+}^{2}\widehat{B}_{-}^{2}+1152E_{2,4}\widehat{A}_{-}\widehat{B}_{-}\widehat{B}_{+}^{2}+1152E_{2,4}\widehat{A}_{+}\widehat{B}_{-}^{2}\widehat{B}_{+}+684E_{2,4}E_{2,2}^{3}\widehat{A}_{-}^{2}\widehat{A}_{+}^{2}\\\ &-1080E_{2,4}^{3}\widehat{A}_{-}\widehat{A}_{+}^{2}\widehat{B}_{-}-1080E_{2,4}^{3}\widehat{A}_{-}^{2}\widehat{A}_{+}\widehat{B}_{+}+1296E_{2,4}^{2}\widehat{A}_{-}^{2}\widehat{B}_{+}^{2}+576E_{2,4}^{2}\widehat{A}_{-}\widehat{A}_{+}\widehat{B}_{-}\widehat{B}_{+}+2952E_{2,2}E_{2,4}^{2}\widehat{A}_{-}\widehat{A}_{+}^{2}\widehat{B}_{-}+2952E_{2,2}E_{2,4}^{2}\widehat{A}_{-}^{2}\widehat{A}_{+}\widehat{B}_{+}-3807E_{2,4}^{4}\widehat{A}_{-}^{2}\widehat{A}_{+}^{2}+7884E_{2,2}E_{2,4}^{3}\widehat{A}_{-}^{2}\widehat{A}_{+}^{2}-4554E_{2,2}^{2}E_{2,4}^{2}\widehat{A}_{-}^{2}\widehat{A}_{+}^{2}\Bigr{)}\\\ &\Bigr{(}16E_{2,2}^{3}\widehat{A}_{+}^{3}\widehat{B}_{+}-1872E_{2,2}^{2}E_{2,4}\widehat{A}_{+}^{3}\widehat{B}_{+}+96E_{2,2}^{2}\widehat{A}_{+}^{2}\widehat{B}_{+}^{2}+2880E_{2,2}E_{2,4}\widehat{A}_{+}^{2}\widehat{B}_{+}^{2}+256E_{2,2}\widehat{A}_{+}\widehat{B}_{+}^{3}-768E_{2,4}\widehat{A}_{+}\widehat{B}_{+}^{3}+E_{2,2}^{4}\widehat{A}_{+}^{4}+636E_{2,2}^{3}E_{2,4}\widehat{A}_{+}^{4}+256\widehat{B}_{+}^{4}-10800E_{2,4}^{3}\widehat{A}_{+}^{3}\widehat{B}_{+}+9072E_{2,2}E_{2,4}^{2}\widehat{A}_{+}^{3}\widehat{B}_{+}\\\ &-6048E_{2,4}^{2}\widehat{A}_{+}^{2}\widehat{B}_{+}^{2}-6399E_{2,4}^{4}\widehat{A}_{+}^{4}+8316E_{2,2}E_{2,4}^{3}\widehat{A}_{+}^{4}-3834E_{2,2}^{2}E_{2,4}^{2}\widehat{A}_{+}^{4}\Bigr{)}\end{array}$
Table 13: Elliptic genera of $(C_{1})_{\pi}$ and $(C_{2})_{\pi}$. Here,
$A_{z}=A_{z}(2\tau)$, $\widehat{A}_{z}=A_{z}(4\tau)$, and same for $B_{z}$.
And $E_{2,2}=E_{2}^{(2)}(\tau),\,E_{2,4}=E_{2}^{(4)}(\tau)$. (The definition
of $\mathcal{I}(\tau,\epsilon_{1},\epsilon_{2})$ can be found in section 4.3.)

$\\{(1,2),2,2\\}$ |  $2j_{+}=$0 | 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---|---
$2j_{-}=$0 |  | 16 |  | 15 |  | 1
1 | 6 |  | 18 |  | 8 |
2 |  | 3 |  | 6 |  | 2
3 | 0 |  | 0 |  | 1 |

$\\{(1,2),2,3\\}$ |  $2j_{+}=$0 | 1 | 2 | 3 | 4
---|---|---|---|---|---
$2j_{-}=$0 | 3 |  | 7 |  | 2
1 |  | 4 |  | 5 |
2 | 0 |  | 1 |  | 1

$\\{(1,2),3,2\\}$ |  $2j_{+}=$0 | 1 | 2 | 3 | 4 | 5 | 6 | 7
---|---|---|---|---|---|---|---|---
$2j_{-}=$0 |  | 97 |  | 98 |  | 17 |  | 0
1 | 52 |  | 129 |  | 73 |  | 7 |
2 |  | 47 |  | 72 |  | 31 |  | 1
3 | 3 |  | 15 |  | 25 |  | 8 |
4 |  | 0 |  | 3 |  | 6 |  | 2
5 | 0 |  | 0 |  | 0 |  | 1 |

$\\{(1,2),3,3\\}$ |  $2j_{+}=$0 | 1 | 2 | 3 | 4 | 5 | 6
---|---|---|---|---|---|---|---
$2j_{-}=$0 | 19 |  | 50 |  | 19 |  | 0
1 |  | 39 |  | 44 |  | 9 |
2 | 7 |  | 22 |  | 19 |  | 2
3 |  | 1 |  | 6 |  | 5 |
4 | 0 |  | 0 |  | 1 |  | 1

$\\{(1,2),4,2\\}$ |  $2j_{+}=$0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
---|---|---|---|---|---|---|---|---|---|---
$2j_{-}=$0 |  | 486 |  | 510 |  | 141 |  | 7 |  | 0
1 | 206 |  | 719 |  | 467 |  | 81 |  | 1 |
2 |  | 343 |  | 490 |  | 245 |  | 28 |  | 0
3 | 54 |  | 179 |  | 229 |  | 96 |  | 7 |
4 |  | 18 |  | 62 |  | 82 |  | 28 |  | 1
5 | 0 |  | 3 |  | 15 |  | 25 |  | 8 |
6 |  | 0 |  | 0 |  | 3 |  | 6 |  | 2
7 | 0 |  | 0 |  | 0 |  | 0 |  | 1 |

$\\{(1,2),4,3\\}$ |  $2j_{+}=$0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8
---|---|---|---|---|---|---|---|---|---
$2j_{-}=$0 | 118 |  | 261 |  | 131 |  | 11 |  | 0
1 |  | 243 |  | 283 |  | 86 |  | 3 |
2 | 65 |  | 183 |  | 155 |  | 34 |  | 0
3 |  | 36 |  | 80 |  | 62 |  | 9 |
4 | 1 |  | 9 |  | 25 |  | 19 |  | 2
5 |  | 0 |  | 1 |  | 6 |  | 5 |
4 | 0 |  | 0 |  | 0 |  | 1 |  | 1

Table 14: Refined GV invariants from 5d $\mathcal{N}=2$ theory of type
$G_{2}$, fiber degree 2, 3 and 4 with mass degree 2 and 3 respectively.

$\\{(1,1,1,1),4,5\\}$ |  $2j_{+}=$0 | 1 | 2
---|---|---|---
$2j_{-}=$0 | 46 |  | 16
1 |  | 35 |
2 | 8 |  | 3

$\\{(1,1,1,1),4,6\\}$ |  $2j_{+}=$0 | 1
---|---|---
$2j_{-}=$0 |  | 7
1 | 5 |

$\\{(1,1,1,1),5,5\\}$ |  $2j_{+}=$0 | 1 | 2 | 3
---|---|---|---|---
$2j_{-}=$0 | 236 |  | 130 |
1 |  | 256 |  | 32
2 | 86 |  | 64 |
1 |  | 16 |  | 4

$\\{(1,1,1,1),5,6\\}$ |  $2j_{+}=$0 | 1 | 2
---|---|---|---
$2j_{-}=$0 |  | 46 |
1 | 38 |  | 14
2 |  | 10 |

$\\{(1,1,1,1),6,5\\}$ |  $2j_{+}=$0 | 1 | 2 | 3 | 4
---|---|---|---|---|---
$2j_{-}=$0 | 1172 |  | 866 |  | 37
1 |  | 1572 |  | 365 |
2 | 634 |  | 595 |  | 48
3 |  | 219 |  | 93 |
4 | 11 |  | 24 |  | 5

$\\{(1,1,1,1),6,6\\}$ |  $2j_{+}=$0 | 1 | 2 | 3
---|---|---|---|---
$2j_{-}=$0 |  | 310 |  | 34
1 | 267 |  | 167 |
2 |  | 128 |  | 21
3 | 13 |  | 15 |

Table 15: Refined GV invariants from 5d $\mathcal{N}=2$ gauge theory of type
$F_{4}$, fiber degree 4, 5 and 6 with mass degree 5 and 6 respectively.

##  Appendix C One-instanton partition functions

In this appendix, we provide the one-instanton partition function
$Z_{1}^{\text{inst}}=\frac{\left(\sqrt{q_{1}}Q_{m}-\sqrt{q_{2}}\right)\left(\sqrt{q_{1}}-\sqrt{q_{2}}Q_{m}\right)}{\left(q_{1}-1\right)\left(q_{2}-1\right)Q_{m}}\tilde{Z}_{1}$
for $G_{2}$, $F_{4}$, $E_{6}$, $E_{7}$, $E_{8}$ with gauge fugacities turned
off. With the notation $v=(q_{1}q_{2})^{-\frac{1}{2}}=e^{-\epsilon_{+}}$, we
observe that $\tilde{Z}_{1,G}$ take the form

$\tilde{Z}_{1,G}=\frac{v^{2h^{\vee}_{G}-2}}{(1-v^{2})^{2h^{\vee}_{G}-2}}\sum_{n=0}^{h^{\vee}_{G}-1}\chi_{(n)}P_{n}(v)\,,$
(161)

with $P_{n}(v)=P_{n}(v^{-1})$. In the massive limit $m\rightarrow\infty$, only
$P_{h^{\vee}_{G}-1}(v)$ survives, and the theory should become the pure gauge
theory with gauge group $G$. We verify that our results here indeed are
reduced to the Hilbert series for pure gauge theories studied in
Benvenuti:2010pq ; Keller:2011ek .

### $G_{2}+1{\text{\bf{Adj}}}$

$\displaystyle\begin{aligned}
\tilde{Z}_{1,G2}&=\frac{v^{6}}{\left(1-v^{2}\right)^{6}}\Big{(}\chi_{(0)}(-7v^{4}+35v^{2}+196+35v^{-2}-7v^{-4})+\chi_{(2)}(15v^{2}+78+15v^{-2})\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(1)}(7v^{3}-133v-133v^{-1}+7v^{-3})+\chi_{(3)}(-{v^{3}}-{8}{v}-8v^{-1}-v^{-3}).\end{aligned}$

### $F_{4}+1{\text{\bf{Adj}}}$

$\displaystyle\begin{aligned}
\tilde{Z}_{1,F4}=\frac{v^{16}}{\left(1-v^{2}\right)^{16}}\Big{(}\chi_{(0)}(-26v^{12}+390v^{10}-1651v^{8}+51948v^{6}\hskip
48.36958pt\\\
-391820v^{4}-5902234v^{2}-11784318-5902234v^{-2}+\cdots-26v^{-16})\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(1)}(26v^{11}-416v^{9}-8762v^{7}-75556v^{5}+3376386v^{3}+14360034v+(v\rightarrow
v^{-1}))\end{aligned}$ $\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(2)}(-273v^{8}+52598v^{6}-690716v^{4}-7548710v^{2}\hskip
133.72786pt\\\ -14516294-7548710v^{-2}+\cdots-273v^{-8})\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(3)}(v^{11}+283v^{9}-7043v^{7}+35983v^{5}+2016456v^{3}+7459088v+(v\rightarrow
v^{-1}))\end{aligned}$ $\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(4)}(-v^{12}-10v^{10}-3v^{8}+4876v^{6}-290771v^{4}\hskip
116.65646pt\quad\quad\quad\\\
-2083666v^{2}-3747250-2083666v^{-2}+\cdots-v^{-12})\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(5)}(26v^{9}-416v^{7}+23842v^{5}+328406v^{3}+1005966v+(v\rightarrow
v^{-1}))\end{aligned}$ $\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(6)}(-1326v^{6}-29731v^{4}-145782v^{2}-240370-145782v^{-2}+\cdots-1326v^{-6})\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(7)}(53v^{7}+1531v^{5}+10897v^{3}+27455v+(v\rightarrow
v^{-1}))\end{aligned}$ $\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(8)}(-v^{8}-36v^{6}-341v^{4}-1208v^{2}-1820-1208v^{-2}+\cdots-v^{-8})\Big{)}.\end{aligned}$

### $E_{6}+1{\text{\bf{Adj}}}$

$\displaystyle\begin{aligned}
\tilde{Z}_{1,E6}=\frac{v^{22}}{\left(1-v^{2}\right)^{22}}\Big{(}\chi_{(0)}(v^{20}-22v^{18}+153v^{16}+98v^{14}-14993v^{12}+158136v^{10}-6926238v^{8}-759340v^{6}\\\
+1523232096v^{4}+7806498348v^{2}+12799549490+7806498348v^{-2}+\cdots+v^{-20})\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(1)}(-v^{21}+22v^{19}-231v^{17}+1618v^{15}-9681v^{13}+64658v^{11}+293625v^{9}\hskip
56.9055pt\\\
+38442635v^{7}-564702589v^{5}-6466371573v^{3}-17713660315v+(v\rightarrow
v^{-1}))\end{aligned}$ $\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(2)}(728v^{14}-15366v^{12}+273910v^{10}-11331190v^{8}+40992458v^{6}+2550510170v^{4}\hskip
28.45274pt\\\
+12242971676v^{2}+19765078892+12242971676v^{-2}+\cdots+728v^{-14})\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(3)}(v^{17}-100v^{15}+1219v^{13}-41216v^{11}+890847v^{9}+17633516v^{7}\hskip
96.73918pt\\\ -538737817v^{5}-4991754986v^{3}-13017447838v+(v\rightarrow
v^{-1}))\end{aligned}$ $\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(4)}(-v^{18}+23v^{16}-175v^{14}+2980v^{12}+64205v^{10}-4291451v^{8}+56601413v^{6}\hskip
48.36958pt\\\
+1259025394v^{4}+5370861892v^{2}+8434831880+5370861892v^{-2}+\cdots-v^{-18})\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(5)}(-v^{17}+22v^{15}-231v^{13}-13683v^{11}+331166v^{9}-1537592v^{7}\hskip
99.58464pt\\\ -200446915v^{5}-1426472661v^{3}-3458377541v+(v\rightarrow
v^{-1}))\end{aligned}$ $\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(6)}(v^{14}+628v^{12}-1771v^{10}-207239v^{8}+20558339v^{6}+247969764v^{4}\hskip
73.97733pt\\\
+913982981v^{2}+1383354810+913982981v^{-2}+\cdots+v^{-14})\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(7)}(-v^{15}+23v^{13}-903v^{11}+16071v^{9}-1430275v^{7}\hskip
162.18062pt\\\ -28606093v^{5}-156881483v^{3}-348520227v+(v\rightarrow
v^{-1}))\end{aligned}$ $\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(8)}(-v^{14}+22v^{12}-231v^{10}+74535v^{8}+2217479v^{6}\hskip
159.3356pt\\\
+17433010v^{4}+55071963v^{2}+79873398+55071963v^{-2}+\cdots-v^{-14})\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(9)}(-3003v^{9}-114959v^{7}-1225510v^{5}-5303376v^{3}-10737012v+(v\rightarrow
v^{-1}))\end{aligned}$ $\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(10)}(79v^{10}+3695v^{8}+50511v^{6}+286946v^{4}+780340v^{2}\hskip
142.26378pt\\\ +1082466+780340v^{-2}+\cdots+79v^{-10})\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(11)}(-v^{11}-56v^{9}-945v^{7}-6776v^{5}-23815v^{3}-43989v+(v\rightarrow
v^{-1}))\Big{)}.\end{aligned}$

### $E_{7}+1{\text{\bf{Adj}}}$

$\displaystyle\begin{aligned}
\tilde{Z}_{1,E7}=\frac{v^{34}}{\left(1-v^{2}\right)^{34}}\Big{(}\chi_{(0)}(&-v^{32}+34v^{30}-561v^{28}+5984v^{26}-47839v^{24}+326535v^{22}-2369840v^{20}+192369v^{18}+319914520v^{16}\\\
&-4555578445v^{14}+45826191560v^{12}+4297712255489v^{10}-38846724860953v^{8}-1018471434602090v^{6}\\\
&-5997290980617505v^{4}-15870630346831724v^{2}-21710965763271066-15870630346831724v^{-2}+\cdots-v^{-32})\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(1)}(&v^{33}-34v^{31}+561v^{29}-5984v^{27}+46243v^{25}-272271v^{23}+1220549v^{21}-84810v^{19}-13375056v^{17}-1414730109v^{15}\\\
&+48668669215v^{13}-1754601580986v^{11}-6682394704228v^{9}+476894906994890v^{7}+4765294386349491v^{5}\\\
&+18118640735313330v^{3}+34067410941177798v+(v\rightarrow
v^{-1}))\end{aligned}$ $\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(2)}(&133v^{26}-4389v^{24}+70091v^{22}-721259v^{20}-12392579v^{18}+511484845v^{16}-10386329537v^{14}+141736436531v^{12}\\\
&+7722900368721v^{10}-93820139003029v^{8}-2070383748708329v^{6}-11777765870556048v^{4}-30742729289953364v^{2}\\\
&-41891685607140374-30742729289953364v^{-2}+\cdots+133v^{-26})\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(3)}(&-133v^{25}+4522v^{23}-65968v^{21}+589722v^{19}+10331706v^{17}-602465756v^{15}+40915328688v^{13}-1908223340828v^{11}\\\
&-24490662408v^{9}+561101153401395v^{7}+5126808165044564v^{5}+18888291604024014v^{3}+35082679743154482v\\\
&+(v\rightarrow v^{-1}))\end{aligned}$ $\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(4)}(&-v^{28}+34v^{26}-561v^{24}-2661v^{22}+389731v^{20}-9787317v^{18}+223445673v^{16}-8739023767v^{14}+167083629053v^{12}\\\
&+4152767497551v^{10}-91661902439757v^{8}-1558312171083212v^{6}-8314456445946167v^{4}-21138018078092692v^{2}\\\
&-28586098919651814-21138018078092692v^{-2}+\cdots-v^{-28})\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(5)}(&v^{29}-34v^{27}+561v^{25}-6117v^{23}+61006v^{21}-847363v^{19}+7090105v^{17}+353568042v^{15}+9836487068v^{13}\\\
&-928207419623v^{11}+6595777319662v^{9}+332856926666326v^{7}+2676443644239847v^{5}+9388310115593702v^{3}\\\
&+17100982922552817v+(v\rightarrow v^{-1}))\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(6)}(&-v^{26}+167v^{24}-6413v^{22}+124354v^{20}-2308041v^{18}+32158356v^{16}-3349371897v^{14}+89887755266v^{12}\\\
&+551143731377v^{10}-49185392946073v^{8}-635558671930867v^{6}-3112506303222448v^{4}-7625247308988574v^{2}\\\
&-10201064010333132-7625247308988574v^{-2}+\cdots-v^{-26})\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(7)}(&v^{27}-34v^{25}+428v^{23}+v^{21}+82818v^{19}-3359334v^{17}+278327436v^{15}-2013396364v^{13}-184969637203v^{11}\\\
&+4802947005758v^{9}+111740083684172v^{7}+775842462654293v^{5}+2556838199829542v^{3}+4539811879506726v\\\
&+(v\rightarrow v^{-1}))\end{aligned}$ $\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(8)}(&-v^{24}+34v^{22}-9206v^{20}+139117v^{18}-6657806v^{16}-431411871v^{14}+19596106410v^{12}-272765774910v^{10}\\\
&-14582938726201v^{8}-146126233882236v^{6}-648220381372136v^{4}-1517814893616390v^{2}-2003377943504808\\\
&-1517814893616390v^{-2}+\cdots-v^{-24})\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(9)}(&v^{25}-34v^{23}+561v^{21}+2661v^{19}-542244v^{17}+46406777v^{15}-978071314v^{13}+4310719436v^{11}+1421475102761v^{9}\\\
&+20892071396056v^{7}+124675707137247v^{5}+382634921027861v^{3}+659189242162231v+(v\rightarrow
v^{-1}))\end{aligned}$ $\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(10)}(&-v^{22}+34v^{20}+41733v^{18}-1754575v^{16}+10060655v^{14}+507682262v^{12}-105235094029v^{10}-2280454391446v^{8}\\\
&-18228754753090v^{6}-72806364194640v^{4}-162029615548868v^{2}-210588997820870-162029615548868v^{-2}\\\
&+\cdots+34v^{-20}-v^{-22})\end{aligned}$ $\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(11)}(&v^{23}-34v^{21}-978v^{19}+4048v^{17}+1238246v^{15}-35782416v^{13}+6125971686v^{11}+191440381209v^{9}+2027213791350v^{7}\\\
&+10437327255044v^{5}+29690319476282v^{3}+49473456685082v+(v\rightarrow
v^{-1}))\end{aligned}$ $\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(12)}(&-v^{20}+1573v^{18}-52887v^{16}+869363v^{14}-293294143v^{12}-12445611684v^{10}-171109984354v^{8}-1120919963028v^{6}\\\
&-4027894422408v^{4}-8493286201536v^{2}-10855716786750-8493286201536v^{-2}+\cdots-v^{-20})\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(13)}(&v^{21}-34v^{19}+561v^{17}-5984v^{15}+11728431v^{13}+626265991v^{11}+10870618606v^{9}+89085511339v^{7}+398918466329v^{5}\\\
&+1050344693949v^{3}+1689966022911v+(v\rightarrow v^{-1}))\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(14)}(&-374395v^{14}-23912735v^{12}-508619265v^{10}-5112652335v^{8}-28093535470v^{6}-91075363610v^{4}-181772936730v^{2}\\\
&-228337756920-181772936730v^{-2}+\cdots-374395v^{-14})\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(15)}(&8778v^{15}+656754v^{13}+16692830v^{11}+201749030v^{9}+1338159284v^{7}+5263002052v^{5}+12840698332v^{3}\\\
&+19939623340v+(v\rightarrow v^{-1}))\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(16)}(&-134v^{16}-11593v^{14}-345521v^{12}-4931707v^{10}-38850151v^{8}-182614170v^{6}-536726278v^{4}-1014596958v^{2}\\\
&-1252490096-1014596958v^{-2}+\cdots-134v^{-16})\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(17)}(v^{17}+99v^{15}+3410v^{13}+56617v^{11}+521917v^{9}+2889898v^{7}+10086066v^{5}+22867856v^{3}+34289476v+(v\rightarrow
v^{-1}))\Big{)}.\end{aligned}$

### $E_{8}+1{\text{\bf{Adj}}}$

$\displaystyle\begin{aligned}
\tilde{Z}_{1,E8}=\frac{v^{58}}{\left(1-v^{2}\right)^{58}}\Big{(}\chi_{(0)}(&-v^{56}+58v^{54}-1653v^{52}+30856v^{50}-424022v^{48}+4567980v^{46}-40079798v^{44}+293578994v^{42}-1824394402v^{40}\\\
&+9667731886v^{38}-42461014998v^{36}+132909329948v^{34}+545475286514v^{32}-18723488678624v^{30}+199657721586v^{28}\\\
&+6070144373847294v^{26}-158359829821935208v^{24}+13123293549284565670v^{22}-304286151278445771080v^{20}\\\
&+93807346004579915840v^{18}+231210520607954818410890v^{16}+212094374121108650515140v^{14}\\\
&-63237430680825110302314085v^{12}-868401895100025356669075258v^{10}-5671384696279393198941920061v^{8}\\\
&-22191130664092299841147741812v^{6}-56604002226248335411717851348v^{4}-98006885388903722205003131524v^{2}\\\
&-117455715138316541428673838364-98006885388903722205003131524v^{-2}+\cdots-v^{-56})\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(1)}(&v^{57}-58v^{55}+1653v^{53}-30856v^{51}+424270v^{49}-4582364v^{47}+40489742v^{45}-301084032v^{43}+1923523102v^{41}\\\
&-10703585571v^{39}+51971657828v^{37}-214401383003v^{35}+599038220846v^{33}-2872323233454v^{31}\\\
&+168035547337370v^{29}-4212687410820234v^{27}+90593403170144332v^{25}-3239694035198676927v^{23}\\\
&-7786725691478936042v^{21}+2727192357545908810241v^{19}-71092705066401885095146v^{17}\\\
&-1219169195760730033562408v^{15}+18317174325496896118960535v^{13}+487044114279693049517489465v^{11}\\\
&+4355779358664347762099988593v^{9}+21602847681572549810258688754v^{7}+67626508813675827650213628830v^{5}\\\
&+141446535661480894042812123181v^{3}+203329332624724715036999923952v+(v\rightarrow
v^{-1}))\end{aligned}$ $\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(2)}(&779247v^{40}-46959451v^{38}+1390356541v^{36}-26958891057v^{34}+1012912518098v^{32}-19606748317983v^{30}\\\
&-64813181649464v^{28}+8085786419270105v^{26}-319990736727661081v^{24}+28351658261313154772v^{22}\\\
&-748943196294632770869v^{20}+1972385457158510035410v^{18}+537227449737270069701604v^{16}\\\
&-173723938370503186697210v^{14}-155088475288876573865779057v^{12}-2058214335347861366919588816v^{10}\\\
&-13255964412188118810033092960v^{8}-51470859163731719012235016574v^{6}-130697683902901841473178066633v^{4}\\\
&-225748482048980748563096528628v^{2}-270339252171845088398941576788-225748482048980748563096528628v^{-2}\\\
&+\cdots-46959451v^{-38}+779247v^{-40})\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(3)}(&1763125v^{39}-100498125v^{37}+3015389375v^{35}-46652194655v^{33}-2473083650543v^{31}+146019040206949v^{29}\\\
&-4267581287915369v^{27}+140382849105058163v^{25}-4643148129781819986v^{23}+17712400328724500496v^{21}\\\
&+3404568482471312364678v^{19}-105871272583204867656060v^{17}-1480793029776444231785780v^{15}\\\
&+29282662577388574143982106v^{13}+695633112048480861279448560v^{11}+6037571298947608844865633794v^{9}\\\
&+29495304460718792477325388428v^{7}+91525807890191640869111761302v^{5}+190469083642574070772294370556v^{3}\\\
&+273160662888213408798843762586v+(v\rightarrow v^{-1}))\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(4)}(&-1763125v^{38}-177215375v^{36}+9162279865v^{34}-208077161810v^{32}+4935311801965v^{30}-58825224897985v^{28}\\\
&-1780691037142485v^{26}-207802905938617496v^{24}+23518066935587317824v^{22}-811213374640556627886v^{20}\\\
&+5533969049770967368644v^{18}+527888804373262377087644v^{16}-1593968004881123694712236v^{14}\\\
&-171243148489213652702326424v^{12}-2133141108451865760603213212v^{10}-13373627029479589437198941070v^{8}\\\
&-51151309281941347159443824298v^{6}-128735881963573152686069567584v^{4}-221297949981119327485353272294v^{2}\\\
&-264607703052426675654332491724-221297949981119327485353272294v^{-2}+\cdots-1763125v^{-38})\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(5)}(&-248v^{45}+14384v^{43}-379564v^{41}+5890248v^{39}+27966805v^{37}-4187360867v^{35}+142822460836v^{33}\\\
&-3496165515746v^{31}+73055252132337v^{29}-2295223662844225v^{27}+121500006713148412v^{25}-3826587262546413180v^{23}\\\
&+49243850310385394258v^{21}+2163057027288332541676v^{19}-93511250346899328197106v^{17}-884856624270726352654586v^{15}\\\
&+29024244939332034225631046v^{13}+590482437162756854448984348v^{11}+4889958518686508618285658630v^{9}\\\
&+23315151466889440009721974542v^{7}+71320669377037428398402581264v^{5}+147198647056877531264934911348v^{3}\\\
&+210297097534691586870217136388v+(v\rightarrow v^{-1}))\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(6)}(&-v^{50}+58v^{48}-1405v^{46}+16720v^{44}-62965v^{42}-703818v^{40}-4466161v^{38}+408946517v^{36}-2713677886v^{34}\\\
&-154668482624v^{32}+4056834767422v^{30}+81550596556093v^{28}-7839674180803077v^{26}-8963349858682126v^{24}\\\
&+9880443699715026855v^{22}-551582734784698348284v^{20}+7021788132815152972097v^{18}+318510994245646293322806v^{16}\\\
&-2547737397547098912719187v^{14}-126768143416544898561539490v^{12}-1451034910174367338520956078v^{10}\\\
&-8755863606942300768953853726v^{8}-32765797942839101287415742906v^{6}-81397403149589411298561278256v^{4}\\\
&-138941704467266571036507101812v^{2}-165762816842880396732666145532-138941704467266571036507101812v^{-2}\\\
&+\cdots-1405v^{-46}+58v^{-48}-v^{-50})\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(7)}(&v^{51}-58v^{49}+1653v^{47}-31104v^{45}+442529v^{43}-5182555v^{41}+52546231v^{39}-435729525v^{37}+1611219988v^{35}\\\
&+36846201591v^{33}-1078632270370v^{31}+20445060694419v^{29}-769965207995803v^{27}+61133860612275391v^{25}\\\
&-2003857524709105545v^{23}+49590149513671515789v^{21}+661473714181831599703v^{19}-55308526729902247488426v^{17}\\\
&-209868346865990507881893v^{15}+20348519237663810642289580v^{13}+349007598222334144825346438v^{11}\\\
&+2721297037268069200651252384v^{9}+12561463335559220201605368670v^{7}+37688495647788439490516123468v^{5}\\\
&+76912917856872391232824496101v^{3}+109307957869017463158752727343v+(v\rightarrow
v^{-1}))\end{aligned}$ $\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(8)}(&-3875v^{42}+224750v^{40}-6405375v^{38}+91392867v^{36}+271599339v^{34}-27610467099v^{32}-1229298897867v^{30}\\\
&+111968835142854v^{28}-5621478945028769v^{26}+73318727870035658v^{24}+928195137349802115v^{22}\\\
&-237448017788272394473v^{20}+5123111892756488763486v^{18}+117755043498443368475248v^{16}\\\
&-2191811579728714517043830v^{14}-67491711160588413066513633v^{12}-700674240481213891236098524v^{10}\\\
&-4032063581415914499690479154v^{8}-14674752126088332714846647544v^{6}-35848940726399344211324636019v^{4}\\\
&-60636997486772361997887074244v^{2}-72132497764650567557687019822-60636997486772361997887074244v^{-2}\\\
&+\cdots+224750v^{-40}-3875v^{-42})\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(9)}(&30380v^{39}+1085v^{37}-50279985v^{35}+2077984095v^{33}+60592436475v^{31}+50736850414v^{29}-115324935643176v^{27}\\\
&+15328472131266604v^{25}-618025661845255476v^{23}+27109498314503455289v^{21}-44072657413395337029v^{19}\\\
&-21558720740026716075154v^{17}+86825411790168279070576v^{15}+10308894943613770056421591v^{13}\\\
&+149234323829450360394565211v^{11}+1085335460390099834467425870v^{9}+4817681814069831288594309950v^{7}\\\
&+14114348349241905710041211705v^{5}+28403320942586731746986059885v^{3}+40103427088449600479655530690v\\\
&+(v\rightarrow v^{-1}))\end{aligned}$ $\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(10)}(&-v^{46}+58v^{44}-1653v^{42}+476v^{40}+1307390v^{38}-45637109v^{36}+745768032v^{34}-7059403107v^{32}-947925309004v^{30}\\\
&+48490585327300v^{28}-1852983003062332v^{26}+48036041789475416v^{24}-1140180289346043561v^{22}\\\
&-57535784401162448555v^{20}+2277608359463371267549v^{18}+19876286601461769607850v^{16}\\\
&-1198250859148174142128764v^{14}-26192752327971055106031243v^{12}-245092924622364383540103494v^{10}\\\
&-1335669910171192925995840246v^{8}-4704118301782910286621534022v^{6}-11263165536466981364560476353v^{4}\\\
&-18842784515328885219283938404v^{2}-22336374768289540026045428446-18842784515328885219283938404v^{-2}\\\
&+\cdots+58v^{-44}-v^{-46})\end{aligned}$ $\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(11)}(&v^{47}-58v^{45}+1653v^{43}-31104v^{41}+469034v^{39}-6754100v^{37}+95895546v^{35}-997979408v^{33}+75277506284v^{31}\\\
&-1850290227551v^{29}+19246197406010v^{27}+316139123756781v^{25}-74934852306761203v^{23}+8418559870497199835v^{21}\\\
&-118755582942355701625v^{19}-4928377489993175546247v^{17}+96442029790213667056394v^{15}\\\
&+3764219423783047365266815v^{13}+46417280428974308965869776v^{11}+313012533488949623853900179v^{9}\\\
&+1329065780351768382842774621v^{7}+3787720466043751517803987162v^{5}+7498286579668805427696488633v^{3}\\\
&+10505832961466043350900914572v+(v\rightarrow v^{-1}))\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(12)}(&-v^{44}+306v^{42}-15789v^{40}+426416v^{38}-5069124v^{36}-46055392v^{34}+907248024v^{32}-170084309061v^{30}\\\
&+8821909613529v^{28}-242492449225431v^{26}+13156242897840703v^{24}-588893170810532845v^{22}\\\
&-3888606133021353960v^{20}+592627891512452388075v^{18}-3505201080890020387158v^{16}-438581357852438167025291v^{14}\\\
&-7366730775368937592398051v^{12}-62041123378346710028242521v^{10}-318628930589129018865328736v^{8}\\\
&-1081537394282008756055096665v^{6}-2530898934808358428736338491v^{4}-4180886987914167423048744943v^{2}\\\
&-4936012096002313577569303188-4180886987914167423048744943v^{-2}+\cdots-v^{-44})\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(13)}(&v^{45}-58v^{43}+1405v^{41}-16472v^{39}-132924v^{37}+9013182v^{35}-152612048v^{33}+11812706298v^{31}-462863160411v^{29}\\\
&+9008566018950v^{27}-759566345512723v^{25}+14103983023370796v^{23}+1320019076000415382v^{21}\\\
&-43411629246390491762v^{19}-323861258952481637926v^{17}+40793635843973108953550v^{15}\\\
&+978923758458393690336321v^{13}+10405497224796024740655624v^{11}+64862930233753203411854027v^{9}\\\
&+262418019157065437267510825v^{7}+725260537783755062737546190v^{5}+1409508299872588791889658839v^{3}\\\
&+1957739036053515467451479934v+(v\rightarrow v^{-1}))\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(14)}(&147250v^{36}-14201131v^{34}+363634464v^{32}-10669131771v^{30}+405750628884v^{28}+17338674957834v^{26}\\\
&+1290929161309556v^{24}-121412129268967574v^{22}+1588014782264264626v^{20}+68798229659053837236v^{18}\\\
&-2951798168594117674774v^{16}-108874375535843145783584v^{14}-1477709033531561671041674v^{12}\\\
&-11218386723070054658290934v^{10}-54131001273632232254719384v^{8}-176538787668409903172388054v^{6}\\\
&-402829555151057086583058422v^{4}-656174683463846496059550902v^{2}-771199039104643325765815292\\\
&-656174683463846496059550902v^{-2}+\cdots+147250v^{-36})\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(15)}(&809627v^{35}-41297735v^{33}+1213201833v^{31}-27410717669v^{29}+361560792904v^{27}-157266482417736v^{25}\\\
&+5880340602342952v^{23}+40234295931501400v^{21}-6388337779469638852v^{19}+158138381108153324756v^{17}\\\
&+10143081400532194815924v^{15}+177878933383196544825692v^{13}+1649447504688955223691112v^{11}\\\
&+9495349309668124610765912v^{9}+36506137778468076110328712v^{7}+97610737580745902905583160v^{5}\\\
&+185924098244625258119111794v^{3}+255784803170473688660432214v+(v\rightarrow
v^{-1}))\end{aligned}$ $\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(16)}(&-v^{40}+58v^{38}-32033v^{36}+983269v^{34}-3684044v^{32}-396326035v^{30}-59167467371v^{28}+8259623614024v^{26}\\\
&-86989527158713v^{24}-9397613052220250v^{22}+371668609354516474v^{20}-5535892927080139732v^{18}\\\
&-794197228916430096890v^{16}-18182394168085282136848v^{14}-206307363525208981322778v^{12}\\\
&-1416494858508468159826154v^{10}-6411527392374038633715810v^{8}-20046703815395749452595942v^{6}\\\
&-44523563448993538410617292v^{4}-71433359422291534252808998v^{2}-83545118232175977334457868\\\
&-71433359422291534252808998v^{-2}+\cdots-v^{-40})\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(17)}(&v^{41}-58v^{39}+1653v^{37}-476v^{35}-1337770v^{33}+45636024v^{31}+4075068473v^{29}-197668659009v^{27}\\\
&-8586968571167v^{25}+616688532587580v^{23}-13270652821774784v^{21}+63256045950755165v^{19}\\\
&+52656758641546779254v^{17}+1582582366587569041730v^{15}+21968958783803540167571v^{13}\\\
&+179676472310670476847988v^{11}+955738113117196477428519v^{9}+3486305416748212061165423v^{7}\\\
&+9002770192592388934208388v^{5}+16784720329113544886566373v^{3}+22856526331356062865650122v\\\
&+(v\rightarrow v^{-1}))\end{aligned}$ $\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(18)}(&-v^{38}+58v^{36}-1653v^{34}+30856v^{32}-146749540v^{30}-3681324859v^{28}+683959984898v^{26}-21386304062010v^{24}\\\
&+186633013567707v^{22}+5438822594379181v^{20}-2995173949943424255v^{18}-117731425214084304706v^{16}\\\
&-1993403023612677922737v^{14}-19370338981413947060273v^{12}-120778219242461321702703v^{10}\\\
&-512615937303211870490020v^{8}-1534465349913942084400545v^{6}-3312824197210386372533605v^{4}\\\
&-5230497883105625529286955v^{2}-6085619763608170162585676-5230497883105625529286955v^{-2}\\\
&+\cdots+58v^{-36}-v^{-38})\end{aligned}$ $\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(19)}(&v^{39}-58v^{37}+1653v^{35}-30856v^{33}+277020v^{31}+456860014v^{29}-22174219192v^{27}+276908622382v^{25}\\\
&+7195630872251v^{23}-371217351686290v^{21}+148787584918035131v^{19}+7517600770497882736v^{17}\\\
&+154222880172226764796v^{15}+1773059946638390321841v^{13}+12915107445197815471466v^{11}\\\
&+63575991814882647792709v^{9}+219883553831912854948949v^{7}+547759467553664243167834v^{5}\\\
&+998657902340137353851409v^{3}+1345383243896925333540604v+(v\rightarrow
v^{-1}))\end{aligned}$ $\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(20)}(&151125v^{32}-13499384v^{30}+217813037v^{28}+8196562378v^{26}-464995140626v^{24}+11558628661480v^{22}\\\
&-6564555474138291v^{20}-413517617757388358v^{18}-10172196515867592250v^{16}-137539898061296235842v^{14}\\\
&-1165237684087911464249v^{12}-6627623812928672491028v^{10}-26389113532247061623200v^{8}\\\
&-75573144625213786098870v^{6}-158462387791614962060855v^{4}-246054973683874545250400v^{2}\\\
&-284733568122818710251734-246054973683874545250400v^{-2}+\cdots+151125v^{-32})\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(21)}(&-3875v^{33}+73625v^{31}+7241259v^{29}-413362897v^{27}+11087994502v^{25}-196982088954v^{23}+258911957331755v^{21}\\\
&+19615662926623491v^{19}+570979585220347732v^{17}+9011118501559084210v^{15}+88314231016254076623v^{13}\\\
&+577832289860731810369v^{11}+2638129161370853741932v^{9}+8651303430483291392248v^{7}\\\
&+20777892854008046163343v^{5}+37021082951919872372783v^{3}+49323667904370057599654v+(v\rightarrow
v^{-1}))\end{aligned}$ $\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(22)}(&-v^{34}+3933v^{32}-226403v^{30}+6436231v^{28}-119991270v^{26}+1648628366v^{24}-9046167202338v^{22}\\\
&-799588990211897v^{20}-27148940843135026v^{18}-495746202005214612v^{16}-5585452053389149006v^{14}\\\
&-41828401731486360745v^{12}-218007820694596785042v^{10}-815242726868999320264v^{8}-2233319018362942312250v^{6}\\\
&-4545943992912889473017v^{4}-6939355426450524863256v^{2}-7985566609619567261206-6939355426450524863256v^{-2}\\\
&+\cdots+3933v^{-32}-v^{-34})\end{aligned}$ $\displaystyle\begin{aligned}
\hskip
59.75095pt+\chi_{(23)}(&v^{35}-58v^{33}+1653v^{31}-30856v^{29}+424270v^{27}-4582116v^{25}+274353541462v^{23}+27736773903360v^{21}\\\
&+1083717132719633v^{19}+22686600167904341v^{17}+291835895897431767v^{15}+2487639612268202035v^{13}\\\
&+14730121539365731508v^{11}+62534852080207394086v^{9}+194558336911726887864v^{7}+450428219780627807278v^{5}\\\
&+784070064521867978771v^{3}+1032867960964930233801v+(v\rightarrow
v^{-1}))\end{aligned}$ $\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(24)}(&-7045404534v^{24}-805014329084v^{22}-35772400374704v^{20}-850811428035964v^{18}-12407651190151244v^{16}\\\
&-119680363804925356v^{14}-801009808071718406v^{12}-3842573046332617736v^{10}-13516588835323464576v^{8}\\\
&-35432252561775870696v^{6}-70006100353818906870v^{4}-105036052816641771420v^{2}\\\
&-120190537433240210820-105036052816641771420v^{-2}+\cdots-7045404534v^{-24})\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(25)}(&148775510v^{25}+19049643406v^{23}+953419225714v^{21}+25548811559972v^{19}+419439941571350v^{17}\\\
&+4550734272855678v^{15}+34244302241654534v^{13}+184726976160586716v^{11}+731286035284214648v^{9}\\\
&+2160725895905789460v^{7}+4823128264826191932v^{5}+8201895969064540290v^{3}+10681972147609674990v+(v\rightarrow
v^{-1}))\end{aligned}$ $\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(26)}(&-2480620v^{26}-353448050v^{24}-19762775652v^{22}-592206494114v^{20}-10872241140216v^{18}-131897019452008v^{16}\\\
&-1109959536372600v^{14}-6699686456861308v^{12}-29708346106549856v^{10}-98485546674356964v^{8}\\\
&-247218326854512912v^{6}-474198468063325920v^{4}-699307885601386920v^{2}-795678443272075320\\\
&-699307885601386920v^{-2}+\cdots-2480620v^{-26})\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(27)}(&30628v^{27}+4827196v^{25}+299475624v^{23}+9968540172v^{21}+203393692040v^{19}+2743366921092v^{17}+25682058074804v^{15}\\\
&+172595935369656v^{13}+853225717878588v^{11}+3158857871953560v^{9}+8875820357073240v^{7}+19112754204541620v^{5}\\\
&+31756812829579200v^{3}+40891682939217780v+(v\rightarrow
v^{-1}))\end{aligned}$ $\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(28)}(&-249v^{28}-43186v^{26}-2955298v^{24}-108632690v^{22}-2449463527v^{20}-36535786736v^{18}-378554380744v^{16}\\\
&-2818846437352v^{14}-15462313229055v^{12}-63637439240298v^{10}-199235067983160v^{8}-479385860055630v^{6}\\\
&-893085571375875v^{4}-1294715203724820v^{2}-1464860072724360-1294715203724820v^{-2}+\cdots-249v^{-28})\end{aligned}$
$\displaystyle\begin{aligned} \hskip
59.75095pt+\chi_{(29)}(&v^{29}+190v^{27}+14269v^{25}+576213v^{23}+14284732v^{21}+234453749v^{19}+2675683550v^{17}+21972715186v^{15}\\\
&+133126452657v^{13}+606326972328v^{11}+2105555153625v^{9}+5634990969615v^{7}+11714759112330v^{5}\\\
&+19025183027595v^{3}+24223919026560v+(v\rightarrow
v^{-1}))\Big{)}.\end{aligned}$

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