We study this family next.

######  Remark 9.1.

The cases of $\Gamma_{0}(p)$ for $p=5$ and $13$ could be treated in a similar
fashion, yielding families of rank $6$ and $14$, respectively.

####  9.2.1 A family associated to $\boldsymbol{\Gamma_{0}(3)}$

We begin by classifying the irreducible congruence representations of
$\Gamma_{0}(3)$. Let $U=S^{-1}T^{-1}S$ and set $A=U^{3}T^{-1}$, so that
$\Gamma_{0}(3)=\big{\langle}T,U^{3}\big{\rangle}=\langle
T,A\rangle=\big{\langle}U^{3},A\big{\rangle}=\Gamma(3)\langle A\rangle$.
Consider the subgroup
$\big{\langle}U^{3},AU^{3}A^{-1},A^{2}U^{3}A^{-2}\big{\rangle}$, which is
normalized by $A$ and contained in $\Gamma(3)$. It follows easily that in fact

$\Gamma(3)=\big{\langle}U^{3},AU^{3}A^{-1},A^{2}U^{3}A^{-2}\big{\rangle}.$

Let $\bar{\Gamma}(3)$ denote the image of $\Gamma(3)$ in
$\operatorname{PSL}_{2}(\mathbb{Z})$. Then the group $\bar{\Gamma}(3)$ is free
of rank $3$, and therefore it is freely generated by the $3$ indicated
elements (after barring them). Now
$\bar{\Gamma}(3)/\bar{\Gamma}(3)^{\prime}\cong\mathbb{Z}^{3}$. Putting these
facts together shows that
$\bar{\Gamma}_{0}(3)/\bar{\Gamma}(3)^{\prime}\cong\mathbb{Z}\wr\mathbb{Z}_{3}$.

######  Lemma 9.2.

We have

$\displaystyle\bar{\Gamma}_{0}(3)/\bar{\Gamma}_{0}(3)^{\prime}\cong\mathbb{Z}\times\mathbb{Z}_{3}$

and the two factors in the quotient are generated by the images of $U^{3}$ and
$A$ respectively.

###### Proof.

The group $\mathbb{Z}\wr C_{3}$ may be realized as $\mathbb{Z}^{3}$ extended
by an order $3$ automorphism that cyclically permutes coordinates. Then we
calculate that the commutator subgroup consists of $(a,b,c)\in\mathbb{Z}^{3}$
with coordinate sum $a+b+c=0$. Therefore the commutator quotient of
$\mathbb{Z}\wr\mathbb{Z}_{3}$ is isomorphic to
$\mathbb{Z}\times\mathbb{Z}_{3}$. The proof follows. ∎

After Lemma 9.2, the characters
$\chi\colon\bar{\Gamma}_{0}(3)\longrightarrow\mathbb{C}^{\times}$ of _finite
order_ are indexed by pairs $(\lambda,\epsilon)$ where $\lambda$ is a root of
unity, and $\epsilon\in\\{0,\pm 1\\}$, and we have

$\displaystyle\chi(U^{3})=\lambda,\qquad\chi(A)=\omega^{\epsilon}.$

For any of the characters $\chi=\chi_{\lambda,\epsilon}$ described in (2.6) we
set

$\rho\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\operatorname{Ind}_{\bar{\Gamma}_{0}(3)}^{\bar{\Gamma}}\chi.$

These are the $4$-dimensional monomial representations of interest. We need a
good left transversal for $\bar{\Gamma}_{0}(3)$ in $\bar{\Gamma}$. We take
left coset representatives to be $I$, $\bar{U}$, $\bar{U}^{2}$, $\bar{S}$, and
choose an ordered basis for the linear space that furnishes $\rho$ to
correspond to the left cosets of $\bar{\Gamma}_{0}(3)$ where we use these
group elements in the indicated order. Name this ordered basis
$(v_{1},v_{2},v_{3},v_{4})$. Then we obtain

$\displaystyle\rho(\bar{U})=\left(\begin{matrix}0&0&\lambda&0\\\ 1&0&0&0\\\
0&1&0&0\\\
0&0&0&\bar{\lambda}\omega^{\epsilon}\end{matrix}\right),\qquad\rho(\bar{S})=\left(\begin{matrix}0&0&0&1\\\
0&0&\omega^{\epsilon}&0\\\ 0&\bar{\omega}^{\epsilon}&0&0\\\
1&0&0&0\end{matrix}\right),$
$\displaystyle\rho(\bar{A})=\left(\begin{matrix}\omega^{\epsilon}&0&0&0\\\
0&0&\lambda\bar{\omega}^{\epsilon}&0\\\
0&0&0&\lambda\bar{\omega}^{\epsilon}\\\
0&\bar{\lambda}^{2}\bar{\omega}^{\epsilon}&0&0\end{matrix}\right).$

We find that

$\displaystyle\rho\big{(}\bar{U}^{3}\big{)}=\operatorname{diag}\big{(}\lambda,\lambda,\lambda,\bar{\lambda}^{3}\big{)},\qquad\rho(\bar{T})=\rho\big{(}\bar{A}^{-1}\bar{U}^{3}\big{)}=\left(\begin{matrix}\lambda\bar{\omega}^{\epsilon}&0&0&0\\\
0&0&0&\bar{\lambda}\omega^{\epsilon}\\\ 0&\omega^{\epsilon}&0&0\\\
0&0&\omega^{\epsilon}&0\end{matrix}\right).$

######  Lemma 9.3.

$\operatorname{im}\rho$ is described by a short exact sequence

$\displaystyle 1\rightarrow K\rightarrow\operatorname{im}\rho\rightarrow
A_{4}\rightarrow 1,$

where
$K\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\rho(\bar{\Gamma}(3))$
is an abelian normal subgroup.

###### Proof.

The group $\bar{\Gamma}(3)$ leaves the span of the first coordinate vector
invariant. Since $\bar{\Gamma}$ acts transitively on the basis then
$\bar{\Gamma}(3)$ acts as a group of diagonal matrices, hence it furnishes an
abelian, normal subgroup of $\operatorname{im}\rho$, call it $K$. Now it
suffices to show that the quotient group by $K$ is isomorphic to $A_{4}$.
Indeed, $\bar{\Gamma}/\bar{\Gamma}(3)\cong A_{4}$, and the nature of the
displayed matrices $\rho\big{(}\bar{U}\big{)}$, $\rho\big{(}\bar{A}\big{)}$,
both of which project onto elements of order $3$, shows that
$\operatorname{im}\rho/K\cong A_{4}$. The Lemma is proved. ∎

######  Lemma 9.4.

$\rho$ is irreducible if, and only if, $\lambda^{4}\neq 1$.

###### Proof.

First assume that $\lambda^{4}\neq 1$, and take an invariant subspace $X$ of
the module furnishing $\rho$. $X$ is spanned by eigenvectors for $U^{3}$. Now,
$X$ cannot contain $v_{4}$ because this generates the full module. Therefore,
$X\subseteq\langle v_{1},v_{2},v_{3}\rangle$. In fact, we claim that
$X\subseteq\langle v_{2},v_{3}\rangle$. Otherwise, some vector in $X$ projects
nontrivially onto $v_{1}$ and then application of $S$ shows that some vector
in $X$ projects nontrivially onto $v_{4}$, an impossibility. Now $U\colon
v_{2}\mapsto v_{3}\mapsto\lambda v_{1}$, and we find that the only possibility
is $X=0$.

Now suppose that $\lambda^{4}=1$. Then $U^{3}$ is represented by the scalar
matrix $\lambda I_{4}$, whence all elements of $\pm\Gamma(3)$ are represented
as scalars, because this group is the normal closure of $\pm U^{3}$ in
$\Gamma$. Then $\operatorname{im}\rho$ is isomorphic to a central extension
(by the group $\langle\lambda\rangle$) of a quotient group of $A_{4}$. This
quotient group must be one of $A_{4}$, $C_{3}$, $1$, and no central extension
of any of these groups has a $4$-dimensional irreducible representation. In
other words, $\rho$ cannot be irreducible if $\lambda^{4}=1$. This completes
the proof of the lemma. ∎

######  Lemma 9.5.

Suppose that $\ker\rho$ contains the principal congruence subgroup
$\bar{\Gamma}(N)$, and suppose that $N$ is the least such integer (the level
of $\ker\rho$). Then $N=2^{a}3^{b}$ for nonnegative integers $a$, $b$.

###### Proof.

We have
$\operatorname{im}\rho\cong\Gamma/\ker\rho\cong(\Gamma/\Gamma(N)/(\ker\rho/\Gamma(N))$.
Suppose that $N=Mp^{k}$ for some prime $p\geq 5$ where $M$ is coprime to $p$
and $k\geq 1$. Then $\Gamma/\Gamma(N)$ has a direct factor isomorphic to
$\Gamma/\Gamma\big{(}p^{k}\big{)}$. This last group is perfect (i.e., it
coincides with its commutator subgroup), and hence it has no nontrivial
solvable quotient groups and in particular $\operatorname{im}\rho$ must be
nonsolvable. This contradicts Lemma 9.3. ∎

######  Example 9.6.

Suppose that $\lambda=1$. In this case we have seen that
$\ker\rho=\bar{\Gamma}(3)$ and $K=1$. In effect, then, $\rho$ is the
representation of $\bar{\Gamma}/\bar{\Gamma}(3)\cong A_{4}$ that obtains if we
induce the $1$-dimensional character of a Sylow $3$-subgroup $A_{4}$ to the
full group.

######  Example 9.7.

We first recall that for each $k\geq 1$ there is a short exact sequence

$\displaystyle
1\rightarrow\bar{\Gamma}(3)/\bar{\Gamma}\big{(}3^{k}\big{)}\rightarrow\bar{\Gamma}/\bar{\Gamma}\big{(}3^{k}\big{)}\rightarrow
A_{4}\rightarrow 1$

and that furthermore
$P\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\Gamma(3)/\Gamma(3^{k})$
is a finite $3$-group. Then a $3$-Sylow-subgroup of the middle group
$\Gamma/\Gamma\big{(}3^{k}\big{)}$ may be taken to be $P\rtimes\langle
A\rangle$. Take a $1$-dimensional character $\chi\colon P\rtimes\langle
A\rangle\rightarrow\mathbb{C}^{\times}$ and let $\chi_{0}$ be the restriction
of $\chi$ to $P$. This character is $\langle A\rangle$-invariant, and we
assume that the stabilizer of $\psi_{0}$ in $A_{4}$ is nothing but $\langle
A\rangle$. In this case we know by Clifford theory, cf. [22, Section 11B],
that the induced representation
$\rho\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\operatorname{Ind}_{P\rtimes\langle
A\rangle}^{\bar{\Gamma}/\bar{\Gamma}(3^{k})}$ is irreducible.

######  Example 9.8.

We have $\left\lvert\bar{\Gamma}(3)/\bar{\Gamma}(9)\right\rvert=3^{3}$. Now
calculate that this group is abelian of exponent $3$, so that
$\bar{\Gamma}(3)/\bar{\Gamma}(9)\cong\mathbb{Z}_{3}^{3}$. Then we can get
several admissible characters $\chi$ with $\lambda$ a primitive cube root of
unity and any $\epsilon$.

######  Lemma 9.9.

For all integers $f\geq 2$ and $M\geq 1$ coprime to $3$, we have
$\bar{\Gamma}\big{(}3^{f}M\big{)}\bar{\Gamma}(3)^{\prime}=\bar{\Gamma}(9)$.

###### Proof.

If $f=2$, $M=1$ this follows from the preceding Example $3$, so we may suppose
that $f\geq 3$. By direct calculation we can find a matrix
$M\in\bar{\Gamma}(3)^{\prime}\setminus{\bar{\Gamma}\big{(}3^{3}\big{)}}$.
Since $1\neq M\bar{\Gamma}\big{(}3^{3}\big{)}\subseteq\bar{\Gamma}(9)$ it
follows that $\bar{\Gamma}\big{(}3^{f}\big{)}\bar{\Gamma}(3)^{\prime}$
projects onto the quotient $\bar{\Gamma}(9)/\bar{\Gamma}\big{(}3^{f}\big{)}$.
This proves the lemma if $M=1$.

If $M>1$ then
$\bar{\Gamma}/\bar{\Gamma}\big{(}3^{f}M\big{)}\cong\bar{\Gamma}/\bar{\Gamma}(M)\times\bar{\Gamma}/\bar{\Gamma}\big{(}3^{f}\big{)}$.
Now apply the previous argument. ∎

######  Corollary 9.10.

In the notation of Lemma 9.5, we always have $b\leq 2$.

######  Lemma 9.11.

In the notation of Lemma 9.5, we always have $a\leq 2$.

###### Proof.

Let the notation be as before and let $K_{0}$ be the odd part of $K$, that is,
the subgroup generated by all elements of odd order. Then $K/K_{0}$ is either
cyclic of order $2^{x}$ or homocyclic $(C_{2^{x}})^{3}$. In the first case our
$A_{4}$ group must act trivially on $K/K_{0}$, and this implies that
$\operatorname{im}\rho$ has a quotient isomorphic to $C_{2^{x-1}}$. Therefore
$x\leq 2$ in this case. In the homocyclic case, we observe that the $A_{4}$
subgroup leaves invariant a $(C_{2^{x}})^{2}$-subgroup of $(C_{2^{x}})^{3}$,
and therefore we may apply the previous argument to the $C_{2^{x}}$-quotient
group to again see that $x\leq 2$. This completes the proof of the lemma. ∎

This finally brings us to

######  Theorem 9.12.

If the induced representation $\rho$ is a congruence representation then its
level must divide $36$.

Now we would like to describe a family of vvmfs corresponding to this data. To
this end, notice that $\Gamma_{0}(3)$ corresponds to the dessins d’enfants

whose associated Belyi function $z$ over the $j$-line – ramified at $0$,
$1728$ and $\infty$ – is defined by the minimal polynomial:

$j=\frac{(z+27)(z+243)^{3}}{z^{3}}.$

The roots of this polynomial generate the function fields of the conjugates of
$\Gamma_{0}(3)$. Newton’s method yields expressions for these conjugates in
terms of $1/j$: if $x=j^{-1/3}$ then the conjugates have expansions

$\displaystyle
z_{1}=j-756-196830j^{-1}-167935356j^{-2}-180552296781j^{-3}-\cdots,$
$\displaystyle
z_{2}=729\big{(}x+12x^{2}+90x^{3}+756x^{4}+8343x^{5}+76788x^{6}+729000x^{7}+835340x^{8}+\cdots\big{)},$

$z_{3}=z_{2}(ux)$ and $z_{4}=z_{2}\big{(}u^{2}x\big{)}$ where $u^{2}+u+1=0$.
This leads to the following $q$-series expansions:

$\displaystyle
z_{1}=\frac{1}{q}-12+54q-76q^{2}-243q^{3}+1188q^{4}-1384q^{5}-2916q^{6}+11934q^{7}+\cdots$
$\displaystyle
z_{2}=729\big{(}q^{1/3}+12q^{2/3}+90q+508q^{4/3}+2391q^{5/3}+9828q^{2}+36428q^{7/3}+\cdots\big{)},$

$z_{3}=z_{2}\big{(}uq^{1/3}\big{)}$ and
$z_{4}=z_{2}\big{(}u^{2}q^{1/3}\big{)}$.

It is well-known that
$z_{1}=\big{(}\eta(q)/\eta\big{(}q^{3}\big{)}\big{)}^{12}$ and
$z_{2}=729\big{(}\eta(q)/\eta\big{(}q^{1/3}\big{)}\big{)}^{12}$. Notice that
if we choose $u={\rm e}^{2\pi{\rm i}/3}$, then $T$ fixes $z_{1}$ and permutes
the other $z_{j}$ as $z_{2}\mapsto z_{3}\mapsto z_{4}\mapsto z_{2}$. From the
transformation law of $\eta$ one finds that $z_{1}|S=z_{2}$. Since $S$ acts on
these functions as a permutation of order $2$, and it does not fix any of them
(otherwise the group leaving them invariant would be too large) we see that
$T$ and $S$ act on the vvmf $(z_{1},z_{2},z_{3},z_{4})^{\rm T}$ via the
matrices

$\displaystyle\rho^{\prime}(T)=\left(\begin{matrix}1&0&0&0\\\ 0&0&1&0\\\
0&0&0&1\\\
0&1&0&0\end{matrix}\right),\qquad\rho^{\prime}(S)=\left(\begin{matrix}0&1&0&0\\\
1&0&0&0\\\ 0&0&0&1\\\ 0&0&1&0\end{matrix}\right).$

The modular derivative $D$ in weight zero is $D=q{\rm d}/{\rm d}q$. Then

$f_{1}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=-\frac{D(z_{1})}{z_{1}}=1+12q+36q^{2}+12q^{3}+84q^{4}+\cdots$

is the unique eigenform of weight $2$ for $\Gamma_{0}(3)$. Said differently,
$z_{1}$ is the unique solution of the MLDE $(D+f_{1})z_{1}=0$. More generally,
let $f_{j}=-D(z_{j})/z_{j}$ and consider the MLDEs $(D+\lambda f_{j})g=0$
which have the solutions $g=z_{j}^{\lambda}$. This $\lambda$ is the exponent
deformation parameter of our family.

Our next goal is to find the induced MLDE of rank $4$ for the full modular
group. To this end, recall that we have $D(E_{4})=-(1/3)E_{6}$,
$D(E_{6})=-(1/2)E_{4}^{2}$ and $D(\Delta)=0$. Thus,

$D(j)=\frac{-E_{4}^{2}E_{6}}{\Delta}=-\frac{E_{6}j}{E_{4}}.$

Differentiating the defining Belyi relation then gives

$f_{j}=\frac{E_{6}(z_{j}+27)(z_{j}+243)}{E_{4}(z^{2}_{j}-486z_{j}-19683)}.$

It follows that $D(f_{j})=-\tfrac{1}{4}f_{j}^{2}+\tfrac{1}{12}E_{4}$ and thus

$\displaystyle D\big{(}z_{j}^{\lambda}\big{)}=-\lambda f_{j}z_{j}^{\lambda},$
$\displaystyle
D^{2}\big{(}z_{j}^{\lambda}\big{)}=\left(\tfrac{4\lambda^{2}+\lambda}{4}f_{j}^{2}-\tfrac{\lambda}{12}E_{4}\right)z_{j}^{\lambda},$
$\displaystyle
D^{3}\big{(}z_{j}^{\lambda}\big{)}=\left(-\tfrac{8\lambda^{3}+6\lambda^{2}+\lambda}{8}f_{j}^{3}+\tfrac{6\lambda^{2}+\lambda}{24}f_{j}E_{4}+\tfrac{\lambda}{36}E_{6}\right)z_{j}^{\lambda},$
$\displaystyle
D^{4}\big{(}z_{j}^{\lambda}\big{)}=\left(\tfrac{32\lambda^{4}+48\lambda^{3}+22\lambda^{2}+3\lambda}{32}f_{j}^{4}-\tfrac{12\lambda^{3}+7\lambda^{2}+\lambda}{24}f_{j}^{2}E_{4}+\tfrac{2\lambda^{2}-\lambda}{96}E_{4}^{2}-\tfrac{8\lambda^{2}+\lambda}{72}f_{j}E_{6}\right)z_{j}^{\lambda}.$

Note that $f_{j}^{4}$, $f_{j}^{2}E_{4}$, $E_{4}^{2}$, $f_{j}E_{6}$ are
linearly dependent:

$f_{j}^{4}=\tfrac{2}{3}f_{j}^{2}E_{4}+\tfrac{1}{27}E_{4}^{2}+\tfrac{8}{27}f_{j}E_{6}.$

Therefore the relation for $D^{4}(z_{j}^{\lambda})$ above can be reexpressed
as

$D^{4}\big{(}z_{j}^{\lambda}\big{)}=\left(\tfrac{32\lambda^{4}+24\lambda^{3}+8\lambda^{2}+\lambda}{48}f_{j}^{2}E_{4}+\tfrac{16\lambda^{4}+24\lambda^{3}+20\lambda^{2}-3\lambda}{432}E_{4}^{2}+\tfrac{64\lambda^{4}+96\lambda^{3}+20\lambda^{2}+3\lambda}{216}f_{j}E_{6}\right)z_{j}^{\lambda}.$

If $z_{j}^{\lambda}$ satisfies a monic MLDE of level one, then we should be
able to find a relation:

$D^{4}\big{(}z_{j}^{\lambda}\big{)}+A(\lambda)E_{4}D^{2}\big{(}z_{j}^{\lambda}\big{)}+B(\lambda)E_{6}D\big{(}z_{j}^{\lambda}\big{)}+C(\lambda)E_{4}^{2}z_{j}^{\lambda}=0$

for rational functions $A,B,C\in\mathbb{Q}(\lambda)$. Since $f_{j}^{2}E_{4}$,
$E_{4}^{2}$ and $f_{j}E_{6}$ are linearly independent, one deduces that

$\displaystyle
A(\lambda)=\tfrac{2}{3}\lambda^{2}+\tfrac{1}{3}\lambda+\tfrac{1}{12},$
$\displaystyle
B(\lambda)=\tfrac{1}{9}\lambda^{3}+\tfrac{4}{9}\lambda^{2}+\tfrac{5}{54}\lambda+\tfrac{1}{72},$
$\displaystyle
C(\lambda)=-\tfrac{1}{27}\lambda\big{(}\lambda-\tfrac{1}{2}\big{)}\big{(}\lambda^{2}+\tfrac{1}{2}\lambda+\tfrac{3}{4}\big{)}.$

This shows that we are in the cyclic case of [39].

Our goal now is to find the monodromy for
$\big{(}z_{1}^{\lambda},z_{2}^{\lambda},z_{3}^{\lambda},z_{4}^{\lambda}\big{)}^{\rm
T}$, which abstractly is isomorphic with the induced representations $\rho$
discussed previously, with $\varepsilon=1$. But since we will want to
symmetrize $\rho^{\prime}(S)$, we will need to determine this representation
exactly within its isomorphism class.

Since $T$ and $S$ permute the differential equations $D+\lambda f_{j}$ in the
same way as they do the $z_{j}$, we find that there are functions of $\lambda$
such that $\Gamma$ acts on the solutions $z_{j}^{\lambda}$ as

$\displaystyle\rho^{\prime}(T)$
$\displaystyle=\left(\begin{matrix}A(\lambda)&0&0&0\\\ 0&0&B(\lambda)&0\\\
0&0&0&C(\lambda)\\\ 0&D(\lambda)&0&0\end{matrix}\right),$
$\displaystyle\rho^{\prime}(S)$
$\displaystyle=\left(\begin{matrix}0&E(\lambda)&0&0\\\ F(\lambda)&0&0&0\\\
0&0&0&G(\lambda)\\\ 0&0&H(\lambda)&0\end{matrix}\right).$

The identity $\rho^{\prime}(S)^{2}=1$ implies that $F=E^{-1}$ and $H=G^{-1}$.
Then the condition that $\rho^{\prime}(R)^{3}=1$ implies that $ABDG=1$ and
$C^{3}=H^{3}$. The monodromy $\rho^{\prime}(T)$ can in fact be read off from
our expansions:

$\displaystyle z_{1}(\tau)^{\lambda}={\rm e}^{-2\pi{\rm
i}\lambda\tau}\prod_{n\geq
0}\left(\frac{1-q^{n}}{1-q^{3n}}\right)^{12\lambda},$ $\displaystyle
z_{2}(\tau)^{\lambda}=729^{\lambda}{\rm e}^{2\pi{\rm
i}\lambda\tau/3}\prod_{n\geq 0}\big{(}1+{\rm e}^{2\pi{\rm i}n\tau/3}+{\rm
e}^{4\pi{\rm i}n\tau/3}\big{)}^{12\lambda},$ $\displaystyle
z_{3}(\tau)^{\lambda}=729^{\lambda}{\rm e}^{2\pi{\rm
i}\lambda(\tau+1)/3}\prod_{n\geq 0}\big{(}1+u{\rm e}^{2\pi{\rm
i}n\tau/3}+u^{2}{\rm e}^{4\pi{\rm i}n\tau/3}\big{)}^{12\lambda},$
$\displaystyle z_{4}(\tau)^{\lambda}=729^{\lambda}{\rm e}^{2\pi{\rm
i}\lambda(\tau+2)/3}\prod_{n\geq 0}\big{(}1+u^{2}{\rm e}^{2\pi{\rm
i}n\tau/3}+u{\rm e}^{4\pi{\rm i}n\tau/3}\big{)}^{12\lambda},$

so that we have

$\displaystyle\rho^{\prime}(T)=\left(\begin{matrix}{\rm e}^{-2\pi{\rm
i}\lambda}&0&0&0\\\ 0&0&1&0\\\ 0&0&0&1\\\ 0&{\rm e}^{2\pi{\rm
i}\lambda}&0&0\end{matrix}\right),\qquad\rho^{\prime}(S)=\left(\begin{matrix}0&E(\lambda)&0&0\\\
E(\lambda)^{-1}&0&0&0\\\ 0&0&0&1\\\ 0&0&1&0\end{matrix}\right).$

The transformation law for $\eta$ under $S$ shows that likewise
$E(\lambda)=1$. Diagonalizing $\rho^{\prime}(T)$ and symmetrising
$\rho^{\prime}(S)$ shows the following:

######  Theorem 9.13.

Let

$F=\frac{1}{\sqrt{3}}\left(\begin{matrix}\sqrt{3}z_{1}^{\lambda}\\\
z_{2}^{\lambda}+u^{-\lambda}z_{3}^{\lambda}+u^{-2\lambda}z_{4}^{\lambda}\\\
z_{2}^{\lambda}+u^{2-\lambda}z_{3}^{\lambda}+u^{1-2\lambda}z_{4}^{\lambda}\\\
z_{2}^{\lambda}+u^{1-\lambda}z_{3}^{\lambda}+u^{2-2\lambda}z_{4}^{\lambda}\end{matrix}\right).$

Then $F$ is a vector-valued modular form of weight $0$ satisfying the
transformation laws

$\displaystyle F(\tau+1)=\left(\begin{matrix}{\rm e}^{-2\pi{\rm
i}\lambda}&0&0&0\\\ 0&{\rm e}^{2\pi{\rm i}\lambda/3}&0&0\\\ 0&0&{\rm
e}^{2\pi{\rm i}(\lambda+1)/3}&0\\\ 0&0&0&{\rm e}^{2\pi{\rm
i}(\lambda+2)/3}\end{matrix}\right)F(\tau),$ $\displaystyle
F(-1/\tau)=\frac{1}{3}\left(\begin{matrix}0&\sqrt{3}&\sqrt{3}&\sqrt{3}\\\
\sqrt{3}&2\cos(2\pi\lambda/3)&2\cos(2\pi(\lambda+2)/3)&2\cos(2\pi(\lambda+1)/3)\\\
\sqrt{3}&2\cos(2\pi(\lambda+2)/3)&2\cos(2\pi(\lambda+1)/3)&2\cos(2\pi\lambda/3)\\\
\sqrt{3}&2\cos(2\pi(\lambda+1)/3)&2\cos(2\pi\lambda/3)&2\cos(2\pi(\lambda+2)/3)\end{matrix}\right)F(\tau).$

In fact, $F$ is a Frobenius family defined in terms of the parameter
$\lambda$.

Let now $\rho$ denote the diagonalized representation as in Theorem 9.13
above, and fix exponents
$L=\operatorname{diag}\big{(}{-}\lambda,\tfrac{\lambda}{3},\tfrac{\lambda+1}{3},\tfrac{\lambda+2}{3}\big{)}$.
If $\lambda$ is not a fourth root of unity then $\rho$ is irreducible by Lemma
9.4, and [39, Theorem 16] shows that $F$ is a form of minimal weight in the
cyclic $D$-module $M(\rho,L)$. That is, $F$, $DF$, $D^{2}F$ and $D^{3}F$
generate $M(\rho,L)$ as $M$-module.

In order to classify characters of VOAs arising in the family $F$ of Theorem
9.13, we’ll need the first few terms in the $q$-expansion of $F$:

$F=\left(\begin{matrix}q^{-\lambda}\big{(}1-12\lambda
q+72\lambda\big{(}\lambda-\tfrac{1}{4}\big{)}q^{2}-288\lambda\big{(}\lambda^{2}-\tfrac{3}{4}\lambda+\tfrac{1}{72}\big{)}q^{3}+\cdots\big{)}\\\
3^{6\lambda+\tfrac{1}{2}}q^{\lambda/3}\big{(}1+288\lambda\big{(}\lambda^{2}+\tfrac{3}{4}\lambda+\tfrac{1}{72}\big{)}q+\cdots\big{)}\\\
3^{6\lambda+\tfrac{1}{2}}q^{(\lambda+1)/3}\big{(}12\lambda+864\lambda\big{(}\lambda^{3}+\tfrac{3}{2}\lambda^{2}+\tfrac{35}{144}\lambda+\tfrac{7}{288}\big{)}q+\cdots\big{)}\\\
3^{6\lambda+\tfrac{1}{2}}q^{(\lambda+2)/3}\big{(}72\lambda\big{(}\lambda+\tfrac{1}{4}\big{)}+\tfrac{10368}{5}\lambda(\lambda+2)\big{(}\lambda+\tfrac{1}{4}\big{)}\big{(}\lambda+\tfrac{1}{6}\big{)}\big{(}\lambda+\tfrac{1}{12}\big{)}q+\cdots\big{)}\end{matrix}\right).$

At this point, since we must preserve the facts that $\rho(T)$ is diagonal and
$\rho(S)$ is real symmetric, we have freedom to permute coordinates of $F$,
swaps signs, and rescale the entire vector by an overall factor. We require in
particular that the coefficients of each coordinate are all of the same sign.
By considering the $q$ coefficient of the first coordinate, this already
forces us to consider $\lambda<0$. By considering the $q$-coefficient of the
second coordinate we find that we likewise require $\lambda>-3/4$.

Further, we need these expansions to be rational. This implies that we must
have $6\lambda+\frac{1}{2}\in\mathbb{Z}$, or that is, $12\lambda\in
1+2\mathbb{Z}$. Therefore, we only have to worry about the values
$\lambda=\tfrac{n}{12}$ for odd $n$ in the range $-8\leq n\leq-1$. When
$n=-3,-6$ the monodromy is reducible, so we in fact only have to consider the
values $\lambda=-\tfrac{1}{12}$, $-\tfrac{5}{12}$ and $-\tfrac{7}{12}$. The
cases $\lambda=-1/12$ and $-7/12$ are accounted for by
$\operatorname{Vir}(c_{2,9})$ and $G_{2,2}$, respectively (after reordering
and rescaling basis vectors appropriately). The case $\lambda=-5/12$ can be
ruled out by noticing that some fusion rules for the corresponding $S$-matrix
are $-1$. Thus, this specialization gives a quasi-conformal modular form that
can’t be re-ordered and scaled to give a conformal modular form. We summarize
this as:

######  Theorem 9.14.

The only specializations of $F$ that give rise to characters of strongly
regular VOAs with irreducible monodromy and exactly $4$ simple modules
$($possibly after rescaling and reordering coordinates of $F)$ are the values
$\lambda=-\tfrac{1}{12}$ and $-\tfrac{7}{12}$, which correspond to
$\operatorname{Vir}(c_{2,9})$ and $G_{2,2}$, respectively.

Once such a family $F$ is computed, it is possible to derive other familes
from it in the search for new character vectors. For example, let now
$G=\eta^{-4}D_{0}F$, so that the exponents have shifted by $-1/6$, and the
$S$-matrix has been replaced by its negative. This is another extremal
Frobenius family where the specializations for generic $\lambda$ live in a
one-dimensional space of vector-valued modular forms (though they are no
longer minimal weight, as the minimal weight has shifted to $k_{1}=-2$).

We find that

$\displaystyle
G=\left(\begin{matrix}q^{-\lambda-\tfrac{1}{6}}\big{(}{-}\lambda+12\lambda\big{(}\lambda-\tfrac{4}{3}\big{)}q-72\lambda\big{(}\lambda-\tfrac{7}{3}\big{)}\big{(}\lambda-\tfrac{7}{12}\big{)}q^{2}+\cdots\big{)}\\\
3^{6\lambda+\tfrac{1}{2}}q^{(2\lambda-1)/6}\big{(}{-}\lambda-288\lambda(\lambda-1)\big{(}\lambda^{3}+\tfrac{3}{4}\lambda+\tfrac{1}{72}\big{)}q+\cdots\big{)}\\\
3^{6\lambda+\tfrac{1}{2}}q^{(2\lambda+1)/6}\big{(}{-}12\lambda^{2}-864\lambda(\lambda-1)\big{(}\lambda^{3}+\tfrac{3}{2}\lambda^{2}+\tfrac{35}{144}\lambda+\tfrac{7}{288}\big{)}q+\cdots\big{)}\\\
3^{6\lambda+\tfrac{1}{2}}q^{(2\lambda+3)/6}\big{(}{-}72\lambda^{2}\big{(}\lambda\\!+\\!\tfrac{1}{4}\big{)}\\!-\\!\tfrac{10368}{5}\lambda(\lambda\\!-\\!1)\big{(}\lambda\\!+\\!\tfrac{1}{12}\big{)}\big{(}\lambda\\!+\\!\tfrac{1}{6}\big{)}\big{(}\lambda\\!+\\!\tfrac{1}{4}\big{)}(\lambda\\!+\\!2)q\\!+\\!\cdots\big{)}\end{matrix}\right)\\!.$

As above, rationality implies that we must have $12\lambda$ equal to an odd
integer. Irreducibility implies that $12\lambda$ is coprime to $3$, so that
$12\lambda\equiv 1,5\pmod{6}$. Next, the fact that all of the Fourier
coefficients in each coordinate of $G$ should have the same sign implies that
if $\lambda>0$ then $\lambda\leq 7/12$, as one sees by examining the first
coordinate. If $\lambda<0$ then examination of the second coordinate implies
that one should have $\lambda>-\tfrac{9}{12}$. That is, we are only interested
in rational $\lambda$ satisfying $\lambda=\tfrac{n}{12}$ where $-7\leq n\leq
7$ and $n\equiv 1,5\pmod{6}$. Consideration of integrality and fusion rules
then implies that only the case $\lambda=-1/12$ could plausibly correspond to
a VOA.

In order to correct fusion rules and signs, one needs to adjust the
specialization of $G$ at $\lambda=-1/12$ by permuting and rescaling as
follows:

$H=\left(\begin{matrix}q^{\tfrac{17}{36}}\big{(}1+25q^{2}+133q^{3}+578q^{4}+1970q^{5}+6076q^{6}+16840q^{7}+\cdots\big{)}\\\
q^{-\tfrac{7}{36}}\big{(}1+13q+98q^{2}+471q^{3}+1780q^{4}+5765q^{5}+16856q^{6}+\cdots\big{)}\\\
q^{\tfrac{5}{36}}\big{(}1+13q+73q^{2}+338q^{3}+1251q^{4}+4048q^{5}+11838q^{6}+\cdots\big{)}\\\
q^{-\tfrac{1}{12}}\big{(}1+17q+116q^{2}+496q^{3}+1817q^{4}+5742q^{5}+16535q^{6}+\cdots\big{)}\end{matrix}\right),$
(9.1)

which has the $S$-matrix:

$\rho(S)=\frac{1}{3}\left(\begin{matrix}2\cos\big{(}\frac{5}{18}\pi\big{)}&-2\cos\big{(}\frac{7}{18}\pi\big{)}&-2\cos\big{(}\frac{1}{18}\pi\big{)}&\sqrt{3}\vspace{1mm}\\\
-2\cos\big{(}\frac{7}{18}\pi\big{)}&-2\cos\big{(}\frac{1}{18}\pi\big{)}&-2\cos\big{(}\frac{5}{18}\pi\big{)}&-\sqrt{3}\vspace{1mm}\\\
-2\cos\big{(}\frac{1}{18}\pi\big{)}&-2\cos\big{(}\frac{5}{18}\pi\big{)}&2\cos\big{(}\frac{7}{18}\pi\big{)}&\sqrt{3}\vspace{1mm}\\\
\sqrt{3}&-\sqrt{3}&\sqrt{3}&0\end{matrix}\right).$

This representation $\rho$ is realized by a modular tensor category, but we do
not know if the conformal modular form $H$ is realized by a strongly regular
VOA.

####  9.2.2 General numerical examples in rank 4

The moduli space of irreducible representations of $\Gamma$ of rank $4$ has
$3$-dimensional components, giving rise to $3$-parameter families of vector-
valued modular forms. It is difficult to handle these $3$-parameter families
systematically as above, where we exploited the special nature of the induced
representations to solve the differential equations of interest. Nevertheless,
one can proceed to search for conformal modular forms arising in the
$3$-parameter families using a numerical approach. Or, in another direction,
one can impose more restrictions such as in [3], to make such computations
more approachable.

We will describe some numerical experiments that we performed in the cyclic
case of [39], and we will assume that $k_{1}=0$, so that
$\operatorname{Tr}(L)=1$. Then, as in ranks $2$ and $3$, up to scaling there
is a unique minimal weight form $F$ that satisfies a monic modular linear
differential equation of the form

$D^{4}F+aE_{4}D^{2}F+bE_{6}DF+cE_{4}^{2}F=0$

for some $a,b,c\in\mathbb{C}$. As shown in [35], this corresponds to the
ordinary differential equation

$\displaystyle\left(\theta_{K}^{4}-\left(\frac{2K+1}{1-K}\right)\theta_{K}^{3}+\left(\frac{44K^{2}-4(9a+7)K+36a+11}{36(1-K)^{2}}\right)\theta_{K}^{2}\right.$
$\displaystyle\left.\qquad{}+\left(\frac{8K^{2}-4(3a+9b+1)K-6a+36b-1}{36(1-K)^{2}}\right)\theta_{K}+\frac{c}{(1-K)^{2}}\right)f=0.$

This equation was already considered in Example 7.9 on page 7.9 above, where
in particular one finds formulas for the unknowns $a$, $b$ and $c$ in terms of
the exponents $e_{1}$, $e_{2}$, $e_{3}$ and $e_{4}$, which must satisfy
$e_{1}+e_{2}+e_{3}+e_{4}=\operatorname{Tr}(L)=1$ thanks to our hypothesis
$k_{1}=0$.

We have searched through the parameter space, focusing on exponents $e_{j}$
corresponding to congruence representations, for conformal specializations. We
succeeded in finding a number of known examples of characters of VOAs, in
addition to a number of as-of-yet undetermined conformal modular forms. The
$S$-matrices in these undetermined cases are:

$S_{1}=\frac{1}{4}\sqrt{1+\frac{1}{\sqrt{5}}}\left(\begin{matrix}\sqrt{5}-1&\sqrt{5}-1&2&2\\\
\sqrt{5}-1&1-\sqrt{5}&2&-2\\\ 2&2&1-\sqrt{5}&1-\sqrt{5}\\\
2&-2&1-\sqrt{5}&\sqrt{5}-1\\\ \end{matrix}\right)$

or

$\displaystyle
S_{2}=\frac{1}{3}\left(\begin{matrix}-\cos\big{(}\tfrac{5\pi}{18}\big{)}\\!+\\!\sqrt{3}\sin\big{(}\tfrac{5\pi}{18}\big{)}\\!&\cos\big{(}\tfrac{5\pi}{18}\big{)}\\!+\\!\sqrt{3}\sin\big{(}\tfrac{5\pi}{18}\big{)}&2\cos\big{(}\tfrac{5\pi}{18}\big{)}&\\!\sqrt{3}\vspace{1mm}\\\
\cos\big{(}\tfrac{5\pi}{18}\big{)}\\!+\\!\sqrt{3}\sin\big{(}\tfrac{5\pi}{18}\big{)}&2\cos\big{(}\tfrac{5\pi}{18}\big{)}&\cos\big{(}\tfrac{5\pi}{18}\big{)}-\sqrt{3}\sin\big{(}\tfrac{5\pi}{18}\big{)}&\\!\\!\\!-\sqrt{3}\\!\vspace{1mm}\\\
2\cos\big{(}\tfrac{5\pi}{18}\big{)}&\\!\cos\big{(}\tfrac{5\pi}{18}\big{)}\\!-\\!\sqrt{3}\sin\big{(}\tfrac{5\pi}{18}\big{)}\\!&-\cos\big{(}\tfrac{5\pi}{18}\big{)}\\!-\\!\sqrt{3}\sin\big{(}\tfrac{5\pi}{18}\big{)}\\!\\!&\\!\\!\sqrt{3}\vspace{1mm}\\\
\sqrt{3}&-\sqrt{3}&\sqrt{3}&0\end{matrix}\right)\\!\\!.$

Note that these two $S$-matrices are realized by known modular tensor
categories. The Fourier coefficients of the corresponding suspected conformal
modular forms are listed in Table 2. The two examples with $\rho(S)=S_{1}$
look like they could perhaps be realized as tensor products of strongly
regular VOAs with two simple modules each, while those with $\rho(S)=S_{2}$
could be harder to identify. We end with a result that helps narrow down the
possibilities of identifying if the first three entries of Table 2 are
realized by VOAs:

$\rho(S)$ | $F_{V}$
---|---
$S_{1}$ | $q^{-\tfrac{33}{40}}\big{(}1+99q+50787q^{2}+2794770q^{3}+70309800q^{4}+1134528021q^{5}+\cdots\big{)}$
|
$q^{\tfrac{17}{40}}\big{(}792+154088q+6610824q^{2}+145807200q^{3}+2162364600q^{4}+\cdots\big{)}$
|
$q^{\tfrac{23}{40}}\big{(}3366+466752q+17581212q^{2}+361184706q^{3}+5110157492q^{4}+\cdots\big{)}$
|
$q^{\tfrac{33}{40}}\big{(}14280+1252152q+39126384q^{2}+721364424q^{3}+9486909432q^{4}+\cdots\big{)}$
$S_{1}$ | $q^{-\tfrac{37}{40}}\big{(}1+37q+65527q^{2}+5306096q^{3}+174479457q^{4}+3487679200q^{5}+\cdots\big{)}$
|
$q^{\tfrac{13}{40}}\big{(}592+223184q+13516544q^{2}+383202192q^{3}+6974809024q^{4}+\cdots\big{)}$
|
$q^{\tfrac{27}{40}}\big{(}11063+1716467q+75169681q^{2}+1783793680q^{3}+28874814615q^{4}+\cdots\big{)}$
|
$q^{\tfrac{37}{40}}\big{(}47840+4779216q+173590384q^{2}+3687784672q^{3}+55362274160q^{4}+\cdots\big{)}$
$S_{2}$ | $q^{\tfrac{-29}{36}}\big{(}1+58q+29319q^{2}+1492282q^{3}+35652194q^{4}+551508428q^{5}+\cdots\big{)}$
|
$q^{\tfrac{31}{36}}\big{(}16588+1295459q+37792162q^{2}+661694421q^{3}+8340292294q^{4}+\cdots\big{)}$
|
$q^{\tfrac{19}{36}}\big{(}1595+230318q+8596093q^{2}+173614474q^{3}+2409567457q^{4}+\cdots\big{)}$
|
$q^{\tfrac{5}{12}}\big{(}1044+195489q+8038422q^{2}+171114471q^{3}+2458828278q^{4}+\cdots\big{)}$
$S_{2}$ | $q^{\tfrac{-29}{36}}\big{(}1+638q+33959q^{2}+1509682q^{3}+35709150q^{4}+551665608q^{5}+\cdots\big{)}$
|
$q^{\tfrac{-5}{36}}\big{(}116+18328q+1302999q^{2}+37817682q^{3}+661768081q^{4}+\cdots\big{)}$
|
$q^{\tfrac{19}{36}}\big{(}1015+228114q+8586233q^{2}+173587214q^{3}+2409491477q^{4}+\cdots\big{)}$
|
$q^{\tfrac{17}{12}}\big{(}190269+8017542q+171051831q^{2}+2458656018q^{3}+26971011288q^{4}+\cdots\big{)}$
Table 2: Suspected conformal modular forms of rank $4$. The exponents $e_{j}$
can be read off from the exponents of the leading $q$-powers.

######  Lemma 9.15.

The first three examples in Table 2 are not a tensor product of a pair of
VOAs, one of which is holomorphic.

###### Proof.

Let $V$ be one of these three examples and assume by way of contradiction that
$V=X\otimes U$, where $U$, $X$ are VOAs and $X$ is holomorphic. Let $L_{V}$ be
the exponent matrix for $V$, for example, for the first entry of Table 2 we
have $L_{V}=\operatorname{diag}(-33/40,17/40,23/40,33/40)$.

If $X$ has central charge $c_{X}$ (a positive integer divisible by $8$) then
the exponent matrix for $U$ is

$L_{U}=\operatorname{diag}(c_{X}/24,c_{X}/24,c_{X}/24,c_{X}/24)+L_{V}.$

The general result that for a strongly regular VOA we have $\tilde{c}>0$ means
that the exponent matrix for such a VOA _cannot_ be nonnegative in the sense
that all entries cannot be nonnegative. Therefore, looking at the first entry
of $L_{U}$, we must have $c_{X}<24$.

Holomorphic VOAs with central charge $<24$ are known: they are lattice
theories $V_{E_{8}}$, $V_{E_{8}\perp E_{8}}$, $V_{\Gamma_{16}}$. In particular
we have $\dim X_{1}=248$ or $496$. Therefore $\dim V_{1}=\dim X_{1}+\dim
U_{1}\geq 248$. But in the three Examples we have $\dim V_{1}=99$, $37$ or
$58$. This contradiction proves the lemma. ∎

### Acknowledgements

Franc was supported by an NSERC Discovery Grant, and Mason was supported by
grant #427007 from the Simons Foundation. We thank these institutions for
their support. We also thank the anonymous referees for their helpful comments
on an earlier draft of this paper.

## References

  * [1] Anderson G., Moore G., Rationality in conformal field theory, Comm. Math. Phys. 117 (1988), 441–450.
  * [2] André Y., Sur la conjecture des $p$-courbures de Grothendieck–Katz et un problème de Dwork, in Geometric Aspects of Dwork Theory, Vols. I, II, Walter de Gruyter, Berlin, 2004, 55–112.
  * [3] Arike Y., Nagatomo K., Sakai Y., Vertex operator algebras, minimal models, and modular linear differential equations of order 4 (with an appendix by Don Zagier), J. Math. Soc. Japan 70 (2018), 1347–1373.
  * [4] Atkin A.O.L., Swinnerton-Dyer H.P.F., Modular forms on noncongruence subgroups, in Combinatorics (Proc. Sympos. Pure Math., Vol. XIX, Univ. California, Los Angeles, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1971, 1–25.
  * [5] Bae J.B., Duan Z., Lee K., Lee S., Sarkis M., Fermionic rational conformal field theories and modular linear differential equations, Prog. Theor. Exp. Phys. 2021 (2021), 08B104, 59 pages, arXiv:2010.12392.
  * [6] Bakalov B., Kirillov Jr. A., Lectures on tensor categories and modular functors, Univ. Lecture Ser., Vol. 21, Amer. Math. Soc., Providence, RI, 2001.
  * [7] Bantay P., The kernel of the modular representation and the Galois action in RCFT, Comm. Math. Phys. 233 (2003), 423–438, arXiv:math.QA/0102149.
  * [8] Bantay P., Modular differential equations for characters of RCFT, J. High Energy Phys. 2010 (2010), no. 6, 021, 17 pages, arXiv:1004.2579.
  * [9] Bantay P., Gannon T., Conformal characters and the modular representation, J. High Energy Phys. 2006 (2006), no. 2, 005, 18 pages, arXiv:hep-th/0512011.
  * [10] Bantay P., Gannon T., Vector-valued modular functions for the modular group and the hypergeometric equation, Commun. Number Theory Phys. 1 (2007), 651–680, arXiv:0705.2467.
  * [11] Bauer M., Coste A., Itzykson C., Ruelle P., Comments on the links between ${\rm su}(3)$ modular invariants, simple factors in the Jacobian of Fermat curves, and rational triangular billiards, J. Geom. Phys. 22 (1997), 134–189, arXiv:hep-th/9604104.
  * [12] Baxter R.J., Hard hexagons: exact solution, J. Phys. A 13 (1980), L61–L70.
  * [13] Borcherds R.E., Vertex algebras, Kac–Moody algebras, and the Monster, Proc. Nat. Acad. Sci. USA 83 (1986), 3068–3071.
  * [14] Calegari F., Dimitrov V., Tang Y., The unbounded denominator conjecture, arXiv:2109.09040.
  * [15] Candelori L., Franc C., Vector-valued modular forms and the modular orbifold of elliptic curves, Int. J. Number Theory 13 (2017), 39–63, arXiv:1506.09192.
  * [16] Cardy J.L., Operator content of two-dimensional conformally invariant theories, Nuclear Phys. B 270 (1986), 186–204.
  * [17] Codogni G., Vertex algebras and Teichmüller modular forms, arXiv:1901.03079.
  * [18] Cohn P.M., Algebra, Vol. 2, Vol. 2, 2nd ed., John Wiley & Sons, Ltd., Chichester, 1989.
  * [19] Conway J.H., Sloane N.J.A., Sphere packings, lattices and groups, 3rd ed., Grundlehren Math. Wiss., Vol. 290, Springer-Verlag, New York, 1999.
  * [20] Creutzig T., Gannon T., Logarithmic conformal field theory, log-modular tensor categories and modular forms, J. Phys. A 50 (2017), 404004, 37 pages, arXiv:1605.04630.
  * [21] Creutzig T., Ridout D., Logarithmic conformal field theory: beyond an introduction, J. Phys. A 46 (2013), 494006, 72 pages, arXiv:1303.0847.
  * [22] Curtis C.W., Reiner I., Representation theory of finite groups and associative algebras, AMS Chelsea Publishing, Providence, RI, 2006.
  * [23] Dong C., Vertex algebras associated with even lattices, J. Algebra 161 (1993), 245–265.
  * [24] Dong C., Li H., Mason G., Twisted representations of vertex operator algebras, Math. Ann. 310 (1998), 571–600, arXiv:q-alg/9509005.
  * [25] Dong C., Li H., Mason G., Modular-invariance of trace functions in orbifold theory and generalized Moonshine, Comm. Math. Phys. 214 (2000), 1–56, arXiv:q-alg/9703016.
  * [26] Dong C., Lin X., Ng S.H., Congruence property in conformal field theory, Algebra Number Theory 9 (2015), 2121–2166, arXiv:1201.6644.
  * [27] Dong C., Mason G., Holomorphic vertex operator algebras of small central charge, Pacific J. Math. 213 (2004), 253–266, arXiv:math.QA/0203005.
  * [28] Dong C., Mason G., Rational vertex operator algebras and the effective central charge, Int. Math. Res. Not. 2004 (2004), 2989–3008, arXiv:math.QA/0201318.
  * [29] Dong C., Mason G., Shifted vertex operator algebras, Math. Proc. Cambridge Philos. Soc. 141 (2006), 67–80, arXiv:math.QA/0411526.
  * [30] Eholzer W., Skoruppa N.P., Conformal characters and theta series, Lett. Math. Phys. 35 (1995), 197–211, arXiv:hep-th/9410077.
  * [31] Eholzer W., Skoruppa N.P., Modular invariance and uniqueness of conformal characters, Comm. Math. Phys. 174 (1995), 117–136, arXiv:hep-th/9407074.
  * [32] Flohr M., Operator product expansion in logarithmic conformal field theory, Nuclear Phys. B 634 (2002), 511–545, arXiv:hep-th/0107242.
  * [33] Flohr M.A.I., Bits and pieces in logarithmic conformal field theory, Internat. J. Modern Phys. A 18 (2003), 4497–4591, arXiv:hep-th/0111228.
  * [34] Franc C., Mason G., Fourier coefficients of vector-valued modular forms of dimension 2, Canad. Math. Bull. 57 (2014), 485–494, arXiv:1304.4288.
  * [35] Franc C., Mason G., Hypergeometric series, modular linear differential equations and vector-valued modular forms, Ramanujan J. 41 (2016), 233–267, arXiv:1503.05519.
  * [36] Franc C., Mason G., Three-dimensional imprimitive representations of the modular group and their associated modular forms, J. Number Theory 160 (2016), 186–214, arXiv:1503.05520.
  * [37] Franc C., Mason G., On the structure of modules of vector-valued modular forms, Ramanujan J. 47 (2018), 117–139, arXiv:1509.07494.
  * [38] Franc C., Mason G., Classification of some vertex operator algebras of rank 3, Algebra Number Theory 14 (2020), 1613–1668, arXiv:1905.07500.
  * [39] Franc C., Mason G., Constructions of vector-valued modular forms of rank four and level one, Int. J. Number Theory 16 (2020), 1111–1152, arXiv:1810.09408.
  * [40] Frenkel I., Lepowsky J., Meurman A., Vertex operator algebras and the Monster, Adv. Pure Appl. Math., Vol. 134, Academic Press, Inc., Boston, MA, 1988.
  * [41] Frenkel I.B., Huang Y.Z., Lepowsky J., On axiomatic approaches to vertex operator algebras and modules, Mem. Amer. Math. Soc. 104 (1993), viii+64 pages.
  * [42] Fuchs J., Schweigert C., Full logarithmic conformal field theory–an attempt at a status report, Fortschr. Phys. 67 (2019), 1910018, 12 pages, arXiv:1903.02838.
  * [43] Gaberdiel M.R., An algebraic approach to logarithmic conformal field theory, Internat. J. Modern Phys. A 18 (2003), 4593–4638, arXiv:hep-th/0111260.
  * [44] Gaberdiel M.R., Hampapura H.R., Mukhi S., Cosets of meromorphic CFTs and modular differential equations, J. High Energy Phys. 2016 (2016), no. 4, 156, 13 pages, arXiv:1602.01022.
  * [45] Gaberdiel M.R., Keller C.A., Modular differential equations and null vectors, J. High Energy Phys. 2008 (2008), no. 9, 079, 29 pages, arXiv:0804.0489.
  * [46] Gainutdinov A.M., A generalization of the Verlinde formula in logarithmic conformal field theory, Theoret. and Math. Phys. 159 (2009), 575–586.
  * [47] Gainutdinov A.M., Jacobsen J.L., Read N., Saleur H., Vasseur R., Logarithmic conformal field theory: a lattice approach, J. Phys. A 46 (2013), 494012, 34 pages, arXiv:1303.2082.
  * [48] Gannon T., The theory of vector-valued modular forms for the modular group, in Conformal Field Theory, Automorphic Forms and Related Topics, Contrib. Math. Comput. Sci., Vol. 8, Springer, Heidelberg, 2014, 247–286, arXiv:1310.4458.
  * [49] Grady J.C., Lam C.H., Tener J.E., Yamauchi H., Classification of extremal vertex operator algebras with two simple modules, J. Math. Phys. 61 (2020), 052302, 19 pages, arXiv:1811.02180.
  * [50] Hampapura H.R., Mukhi S., Two-dimensional RCFT’s without Kac–Moody symmetry, J. High Energy Phys. 2016 (2016), no. 7, 138, 19 pages, arXiv:1605.03314.
  * [51] Hecke E., Mathematische Werke, 3rd ed., Vandenhoeck & Ruprecht, Göttingen, 1983\.
  * [52] Hirzebruch F., Berger T., Jung R., Manifolds and modular forms, Aspects of Mathematics, Vol. E20, Friedr. Vieweg & Sohn, Braunschweig, 1992.
  * [53] Höhn G., Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Bonner Mathematische Schriften, Vol. 286, Universität Bonn, Mathematisches Institut, Bonn, 1996.
  * [54] Huang Y.Z., Vertex operator algebras and the Verlinde conjecture, Commun. Contemp. Math. 10 (2008), 103–154, arXiv:math.QA/0406291.
  * [55] Iwasaki K., Kimura H., Shimomura S., Yoshida M., From Gauss to Painlevé. A modern theory of special functions, Aspects of Mathematics, Vol. E16, Friedr. Vieweg & Sohn, Braunschweig, 1991.
  * [56] Kaidi J., Lin Y.H., Parra-Martinez J., Holomorphic modular bootstrap revisited, J. High Energy Phys. 2021 (2021), no. 12, 151, 48 pages, arXiv:2107.13557.
  * [57] Kawai S., Wheater J.F., Modular transformation and boundary states in logarithmic conformal field theory, Phys. Lett. B 508 (2001), 203–210, arXiv:hep-th/0103197.
  * [58] Knopp M., Mason G., Vector-valued modular forms and Poincaré series, Illinois J. Math. 48 (2004), 1345–1366.
  * [59] Lam C.H., Shimakura H., 71 holomorphic vertex operator algebras of central charge 24, Bull. Inst. Math. Acad. Sin. (N.S.) 14 (2019), 87–118.
  * [60] Li H.S., Symmetric invariant bilinear forms on vertex operator algebras, J. Pure Appl. Algebra 96 (1994), 279–297.
  * [61] Marks C., Irreducible vector-valued modular forms of dimension less than six, Illinois J. Math. 55 (2011), 1267–1297, arXiv:1004.3019.
  * [62] Marks C., Mason G., Structure of the module of vector-valued modular forms, J. Lond. Math. Soc. 82 (2010), 32–48, arXiv:0901.4367.
  * [63] Mason G., Lattice subalgebras of strongly regular vertex operator algebras, in Conformal Field Theory, Automorphic Forms and Related Topics, Contrib. Math. Comput. Sci., Vol. 8, Springer, Heidelberg, 2014, 31–53, arXiv:1110.0544.
  * [64] Mason G., Vertex rings and their Pierce bundles, in Vertex Algebras and Geometry, Contemp. Math., Vol. 711, Amer. Math. Soc., Providence, RI, 2018, 45–104, arXiv:1707.00328.
  * [65] Mason G., Five not-so-easy pieces: open problems about vertex rings, in Vertex Operator Algebras, Number Theory and Related Topics, Contemp. Math., Vol. 753, Amer. Math. Soc., Providence, RI, 2020, 213–232, arXiv:1812.06206.
  * [66] Mason G., Nagatomo K., Sakai Y., Vertex operator algebras of rank 2 – the Mathur–Mukhi–Sen theorem revisited, Commun. Number Theory Phys. 15 (2021), 59–90.
  * [67] Mason G., Tuite M., Vertex operators and modular forms, in A Window Into Zeta and Modular Physics, Math. Sci. Res. Inst. Publ., Vol. 57, Cambridge University Press, Cambridge, 2010, 183–278, arXiv:0909.4460.
  * [68] Mathur S.D., Mukhi S., Sen A., On the classification of rational conformal field theories, Phys. Lett. B 213 (1988), 303–308.
  * [69] Mathur S.D., Mukhi S., Sen A., Reconstruction of conformal field theories from modular geometry on the torus, Nuclear Phys. B 318 (1989), 483–540.
  * [70] Nagatomo K., Mason G., Sakai Y., Vertex operator algebras with central charge 8 and 16, in Vertex Operator Algebras, Number Theory and Related Topics, Contemp. Math., Vol. 753, Amer. Math. Soc., Providence, RI, 2020, 157–186, arXiv:1812.06357.
  * [71] Nagi J., Operator algebra in logarithmic conformal field theory, Phys. Rev. D 72 (2005), 086004, 9 pages, arXiv:hep-th/0507242.
  * [72] Ng S.H., Wang Y., Wilson S., On symmetric representations of $\mathbf{SL}_{2}(\mathbf{Z})$, arXiv:2203.15701.
  * [73] Niemeier H.V., Definite quadratische Formen der Dimension $24$ und Diskriminante $1$, J. Number Theory 5 (1973), 142–178.
  * [74] Rankin R.A., Modular forms and functions, Cambridge University Press, Cambridge – New York – Melbourne, 1977.
  * [75] Sabbah C., Isomonodromic deformations and Frobenius manifolds. An introduction, Universitext, Springer-Verlag London, Ltd., London, 2007.
  * [76] Schellekens A.N., Meromorphic $c=24$ conformal field theories, Comm. Math. Phys. 153 (1993), 159–185, arXiv:hep-th/9205072.
  * [77] Serre J.P., A course in arithmetic, Grad. Texts in Math., Vol. 7, Springer-Verlag, New York – Heidelberg, 1973.
  * [78] Sloane N.J.A., The on-line encyclopedia of integer sequences, 2020, http://oeis.org/.
  * [79] Tuba I., Wenzl H., Representations of the braid group $B_{3}$ and of ${\rm SL}(2,{\bf Z})$, Pacific J. Math. 197 (2001), 491–510, arXiv:math.RT/9912013.
  * [80] Vidunas R., Dihedral Gauss hypergeometric functions, Kyushu J. Math. 65 (2011), 141–167, arXiv:0807.4888.
  * [81] Zagier D., Elliptic modular forms and their applications, in The 1-2-3 of Modular Forms, Universitext, Springer, Berlin, 2008, 1–103.
  * [82] Zhu Y., Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 (1996), 237–302.