# Dimension formulas for spaces of vector-valued Siegel modular forms of
degree two and level two

Jonas Bergström and Fabien Cléry<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Using a description of the cohomology of local systems on the moduli space of
abelian surfaces with a full level two structure, together with a computation
of Euler characteristics we find the isotypical decomposition, under the
symmetric group on 6 letters, of spaces of vector-valued Siegel modular forms
of degree two and level two.

##  1\. Introduction

In this paper, we refine the previously known dimension formulas for spaces
$M_{k,j}(\Gamma[2])$ of vector-valued Siegel modular forms of degree 2 and
level 2, by determining their isotypical decomposition under the action of
${\mathrm{Sp}}(4,\mathbb{F}_{2})\cong\mathfrak{S}_{6}$. This extends previous
work, see for instance [18, 1, 28, 29, 16, 17, 30, 12]. In particular,
Tsushima gave in [29, Theorems 2, 3] a formula for the dimension of the space
$S_{k,j}(\Gamma[N])$ for any $N$ under the conditions $j\geqslant 1$ and
$k\geqslant 5$ or $j=0$ and $k\geqslant 4$. The ranges for $j$ and $k$ in
Tsushima’s dimension formula for $N=2$ have been slightly extended in [12,
Theorem 12.1]. There is an overview of dimension formulas such as these on the
webpage [26].

In [4], there is a conjectural description of the motivic Euler
characteristic, with its isotypical decomposition under the action of
$\mathfrak{S}_{6}$, of any symplectic local system on $\mathcal{A}_{2}[2]$ the
moduli space of abelian surfaces with a full level two structure. These
conjectures were later proven in [23]. In particular, this gives us the
integer-valued Euler characteristic of an isotypical component under
$\mathfrak{S}_{6}$ of any local system on $\mathcal{A}_{2}[2]$ as a sum of a
well-known value (in terms of dimensions of spaces of elliptic modular cusp
forms) plus four times the dimension of the (isotypical component of the)
vector space of Siegel modular cusp forms of degree 2 and level 2, see Theorem
5.3. In Section 4 we then find an effective formula to compute these integer-
valued Euler characteristics. This is achieved by stratifying the moduli space
$\mathcal{A}_{2}[2]$ in terms of the automorphism groups of principally
polarized abelian surfaces, which are Jacobians of smooth projective curves of
genus $2$, or products of elliptic curves. By computing the action of these
automorphism groups and of $\mathfrak{S}_{6}$, on the first cohomology group
of the corresponding abelian surfaces, we can find a formula for the integer-
valued Euler characteristic, see Equation (9). This is a method previously
used for instance in [15].

In Section 2 we give an overview of the Siegel modular forms we are interested
in, together with a short description of the Arthur packets for
$\mathrm{GSp}(4)$. Then, in Section 3 and Section 5 we include isotypical
decompositions of the spaces of Siegel modular forms of degree $2$ and level
$2$ to give a comprehensive reference for these results. Computer programs,
written in Sage, which compute all results of this paper, are provided on a
GitHub repository [3]. Tables with some of the results of this paper can also
be found on the webpage [6].

##  2\. Siegel modular forms

The level 2 congruence subgroups we are concerned with are the following ones

$\Gamma[2]=\left\\{\gamma\in\Gamma:\gamma\equiv 1_{4}\bmod
2\right\\},\quad\Gamma_{1}[2]=\left\\{\gamma\in\Gamma:\gamma\equiv\left(\begin{smallmatrix}1_{2}&*\\\
0&1_{2}\end{smallmatrix}\right)\bmod
2\right\\},\quad\Gamma_{0}[2]=\left\\{\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\Gamma:c\equiv 0\bmod 2\right\\},$

where
$\Gamma={\mathrm{Sp}}(4,\mathbb{Z})=\\{\gamma\in{\mathrm{GL}}(4,\mathbb{Z}):\gamma^{t}J\gamma=J\\}$
with $J=\left(\begin{smallmatrix}0&1_{2}\\\ -1_{2}&0\end{smallmatrix}\right)$
and $1_{n}$ the identity matrix of size $n$. We clearly have the following
inclusions $\Gamma[2]<\Gamma_{1}[2]<\Gamma_{0}[2]<\Gamma$ and the successive
quotients can be identified as follows

$\Gamma_{1}[2]/\Gamma[2]\cong(\mathbb{Z}/2\mathbb{Z})^{3},\quad\Gamma_{0}[2]/\Gamma[2]\cong\mathbb{Z}/2\mathbb{Z}\times\mathfrak{S}_{4},\quad\Gamma_{0}[2]/\Gamma_{1}[2]\cong\mathfrak{S}_{3},\quad\Gamma/\Gamma[2]\cong\mathfrak{S}_{6},$

with $\mathfrak{S}_{n}$ the symmetric group on $n$ letters. As usual, theses
groups act on the Siegel upper half space $\mathfrak{H}_{2}$ of degree 2:
$\mathfrak{H}_{2}=\\{\tau\in\text{Mat}(2\times
2,\mathbb{C}):\tau^{t}=\tau,\text{Im}(\tau)>0\\}$ via
$\tau\mapsto\gamma\,\tau=(a\tau+b)(c\tau+d)^{-1}$. More details on the
orbifolds of the action of the previous groups can be found in [12, Section
2]. We let $G$ be one of the groups $\Gamma[2],\Gamma_{1}[2],\Gamma_{0}[2]$ or
$\Gamma$. For any integer $k$ and non-negative integer $j$, the space of
modular forms of weight $(k,j)$ on $G$ is denoted by $M_{k,j}(G)$ and is
defined by

$\displaystyle
M_{k,j}(G)=\\{f\colon\mathfrak{H}_{2}\to\mathbb{C}^{j+1}\,\text{holomorphic}\,|\,$
$\displaystyle
f((a\tau+b)(c\tau+d)^{-1})=\text{Sym}^{j}(c\tau+d)\otimes\det(c\tau+d)^{k}f(\tau)\,\text{
for all }\,\gamma=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in G\\}.$

The subspace of cusp forms of $M_{k,j}(G)$ will be denoted by $S_{k,j}(G)$,
this is the kernel of the (global) Siegel $\Phi$-operator. Let us make a
couple of easily verified remarks. Firstly, since $-1_{4}$ belongs to the
group $\Gamma[2]$, we have that $M_{k,j}(\Gamma[2])=\\{0\\}$ if $j$ is odd.
This can be directly read off from the functional equation satisfied by any
element of $M_{k,j}(\Gamma[2])$. Therefore, from now on, we assume that $j$ is
even. Secondly, if $k$ is odd then $M_{k,j}(\Gamma[2])=S_{k,j}(\Gamma[2])$:
let $\Gamma(2)$ be the principal congruence subgroup of level $2$ of
${\mathrm{SL}}(2,\mathbb{Z})$, the (global) Siegel $\Phi$-operator maps
$M_{k,j}(\Gamma[2])$ to $M_{j+k}(\Gamma(2))^{\oplus 15}$ (note $15$ is the
number of $1$-dimensional cusps of the group $\Gamma[2]$) and since $-1_{2}$
belongs to $\Gamma(2)$, the space $M_{j+k}(\Gamma(2))$ reduces to ${0}$ when
$k$ is odd ($j$ is even). These two facts also hold for the groups
$\Gamma_{1}[2],\Gamma_{0}[2]$ and $\Gamma$. A less easy fact is the
generalisation of the Koecher principle to vector-valued Siegel modular forms,
see [13, Satz 1], which implies that

$M_{k,j}(\Gamma[2])=\left\\{0\right\\}\text{ for any }k<0\text{ for any }j.$

The Petersson inner product provides an orthogonal decomposition

$M_{k,j}(G)=E_{k,j}(G)\oplus S_{k,j}(G).$ (1)

We call $E_{k,j}(G)$ the space of Eisenstein series. The decomposition (1) can
be refined according to the classification of automorphic representations of
G${\mathrm{Sp}}(4)$: Arthur packets. There are six different types of Arthur
packets (see [2], or [25, pp. 3088–3089]):

  * •

(G): general type. They can only appear in $S_{k,j}(G)$.

  * •

(Y): Yoshida type. They can only appear in $S_{k,j}(G)$. Modular forms of this
type are also called Yoshida lifts.

  * •

(Q): Soudry type (Klingen parabolic). They can only appear in $E_{k,j}(G)$.
Modular forms of this type are also called Klingen-Eisenstein series.

  * •

(P): Saito-Kurokawa type (Siegel parabolic). They can only appear in
$S_{k,0}(G)$, i.e. they are scalar-valued cusp forms. Modular forms of this
type are also called Saito-Kurokawa lifts.

  * •

(B): Howe–Piatetski-Shapiro type (Borel parabolic). They can only appear in
the space of cusp forms, but they do not appear in $S_{k,j}(G)$.

  * •

(F): Finite type. They can only appear in $E_{k,j}(G)$. Modular forms of this
type are also called Siegel-Eisenstein series.

So we have

$E_{k,j}(G)=E^{\textbf{(F)}}_{k,j}(G)\oplus
E^{\textbf{(Q)}}_{k,j}(G)\quad\text{and}\quad
S_{k,j}(G)=S^{\textbf{(G)}}_{k,j}(G)\oplus S^{\textbf{(P)}}_{k,j}(G)\oplus
S^{\textbf{(Y)}}_{k,j}(G).$ (2)

Since the group $\Gamma[2]$ is a normal subgroup of $\Gamma$ (it is the kernel
of the reduction modulo $2$), we get an action of $\Gamma$ on the space
$M_{k,j}(\Gamma[2])$

$\begin{smallmatrix}{\mathrm{Sp}}(4,\mathbb{Z})\times
M_{k,j}(\Gamma[2])&\to&M_{k,j}(\Gamma[2])\\\
(\gamma,f)&\mapsto&f|_{k,j}\gamma^{-1}\end{smallmatrix}$

From this action, we deduce a group homomorphism

$\begin{smallmatrix}\Gamma&\to&{\mathrm{GL}}(M_{k,j}(\Gamma[2]))\\\
\gamma&\mapsto&\left(\begin{smallmatrix}M_{k,j}(\Gamma[2])&\to&M_{k,j}(\Gamma[2])\\\
f&\mapsto&f|_{k,j}\gamma^{-1}\end{smallmatrix}\right)\end{smallmatrix}$

whose kernel obviously contains the group $\Gamma[2]$. So the previous
homomorphism factors through the group $\Gamma[2]$ and we obtain a group
homomorphism

$\Gamma/\Gamma[2]\cong{\mathrm{Sp}}(4,\mathbb{F}_{2})\cong\mathfrak{S}_{6}\to{\mathrm{GL}}(M_{k,j}(\Gamma[2]))$

i.e. a representation of the group $\mathfrak{S}_{6}$ on the space
$M_{k,j}(\Gamma[2])$. Note that the second isomorphism is ambiguous due to the
outer automorphism of $\mathfrak{S}_{6}$ so we need to fix this isomorphism.
We fix this isomorphism as follows:
$\mathfrak{S}_{6}=\langle(12),(123456)\rangle$ and as in [12, Equation (3.2)]
(see also [18, pp. 398-399]), we set

$(12)\mapsto\left(\begin{smallmatrix}1&0&1&0\\\ 0&1&0&0\\\ 0&0&1&0\\\
0&0&0&1\end{smallmatrix}\right)\bmod
2\quad\text{and}\quad(123456)\mapsto\left(\begin{smallmatrix}0&1&0&1\\\
1&0&1&0\\\ 1&0&1&1\\\ -1&1&0&1\end{smallmatrix}\right)\bmod 2.$

The irreducible representations of $\mathfrak{S}_{n}$ correspond bijectively
with the partitions of $n$. The representation of the symmetric group
$\mathfrak{S}_{n}$ corresponding with the partition $\varpi$ will be denoted
by $s[\varpi]$, with $s[n]$ the trivial one and $s[1^{n}]$ the alternating
one.

Table 1. Irreducible representations of $\mathfrak{S}_{6}$ and their dimensions $[\varpi]$ | $[6]$ | $[5,1]$ | $[4,2]$ | $[4,1^{2}]$ | $[3^{2}]$ | $[3,2,1]$ | $[3,1^{3}]$ | $[2^{3}]$ | $[2^{2},1^{2}]$ | $[2,1^{4}]$ | $[1^{6}]$
---|---|---|---|---|---|---|---|---|---|---|---
$\dim s[\varpi]$ | 1 | 5 | 9 | 10 | 5 | 16 | 10 | 5 | 9 | 5 | 1

For a representation $V$ of the symmetric group $\mathfrak{S}_{n}$ we write

$\dim_{\mathfrak{S}_{n}}V=\sum_{\varpi}m_{s[\varpi]}(V)\cdot
s[\varpi]\in\mathbb{Z}[\mathfrak{S}_{n}],$ (3)

where $\mathbb{Z}[\mathfrak{S}_{n}]$ is the representation ring and
$m_{\varpi}(V)$ is the multiplicity of the representation $s[\varpi]$
appearing in $V$. We call the right hand side of (3) the isotypical
decomposition of $V$. Knowing the isotypical decomposition of a space
$M_{k,j}(\Gamma[2])$ gives us all the information we want about the spaces
$M_{k,j}(\Gamma_{1}[2])$, $M_{k,j}(\Gamma_{0}[2])$ and $M_{k,j}(\Gamma)$ by
representation theory: to an isotypical decomposition
$\dim_{\mathfrak{S_{6}}}M_{k,j}(\Gamma[2])=m_{s[6]}\,s[6]+m_{s[5,1]}\,s[5,1]+\cdots+m_{s[1^{6}]}\,s[1^{6}]$
contributes

$\displaystyle\dim_{\mathfrak{S_{3}}}M_{k,j}(\Gamma_{1}[2])=(m_{s[6]}+m_{s[4,2]}+m_{s[2^{3}]})s[3]+(m_{s[5,1]}+m_{s[4,2]}+m_{s[3,2,1]})s[2,1]+(m_{s[4,1^{2}]}+m_{s[3^{2}]})s[1^{3}],$
$\displaystyle\dim M_{k,j}(\Gamma_{0}[2])=m_{s[6]}+m_{s[4,2]}+m_{s[2^{3}]},$
$\displaystyle\dim M_{k,j}(\Gamma)=\dim M_{k,j}(\Gamma[2])^{s[6]}=m_{s[6]},$
$\displaystyle\dim M_{k,j}(\Gamma,\varepsilon)=\dim
M_{k,j}(\Gamma[2])^{s[1^{6}]}=m_{s[1^{6}]}.$

Here, $\epsilon$ denotes the unique non-trivial character of $\Gamma$, see
[10, Section 12] for a brief description of this character. Therefore we focus
on the spaces $M_{k,j}(\Gamma[2])$ in the sequel. In the case of scalar-valued
modular forms, the previous decompositions allow us to recover the results
given in [24, Appendix A] for the groups $\Gamma,\Gamma_{0}[2]$ and
$\Gamma[2]$. In the case of scalar-valued modular forms we write
$M_{k}(\Gamma[2])=M_{k,0}(\Gamma[2])$ and
$S_{k}(\Gamma[2])=S_{k,0}(\Gamma[2])$.

##  3\. Isotypical decompositions in the scalar-valued case

By the Koecher principle, we know that $M_{k}(\Gamma[2])=\\{0\\}$ for $k<0$.
The following theorem is due to Igusa, see [18, p.398].

######  Theorem 3.1 (Igusa).

We have

$\dim M_{k}(\Gamma[2])=\begin{cases}\frac{(k+1)(k^{2}+2k+12)}{12}\hskip
20.0pt\text{if}\,\,\,k\geqslant 0\,\,\,even\\\ \dim M_{k-5}(\Gamma[2])\hskip
13.0pt\text{if}\,\,\,\,k\geqslant 1\,\,\,\text{odd}.\end{cases}$

For $k$ odd, the last equality comes from
$M_{k}(\Gamma[2])=S_{k}(\Gamma[2])=\chi_{5}\cdot M_{k-5}(\Gamma[2])$ where
$\chi_{5}$ denotes the unique cusp form, up to a multiplicative constant,
generating the space $S_{5}(\Gamma[2])$. In fact, Igusa did more than
computing the dimension of the spaces $M_{k}(\Gamma[2])$. He also computed the
characters of $\mathfrak{S}_{6}$ on the space $M_{k}(\Gamma[2])$ (see [18,
Theorem 2]) and he showed that as $\mathfrak{S}_{6}$-representation for $k$
even we have

$\dim_{\mathfrak{S}_{6}}M_{k}(\Gamma[2])={\text{Sym}}^{k/2}(s[2^{3}])-\begin{cases}0\hskip
78.24507pt\text{ if }k\in\\{0,2,4,6\\}\\\
{\text{Sym}}^{k/2-4}(s[2^{3}])\,\,\,\,\,\,\,\,\,\,\,\,\text{ if }k\geqslant
8,\end{cases}$

where we put ${\mathrm{Sym}}^{0}(s[\varpi])=s[n]$ for any irreducible
representation $s[\varpi]$ of $\mathfrak{S}_{n}$. Note that the relation
appearing in weight $8$ defines the Igusa quartic. From the results of Igusa,
we deduce the generating series for the multiplicity of the irreducible
representations of $\mathfrak{S}_{6}$ in $M_{k}(\Gamma[2])$:

$s[\varpi]$ | $\sum_{k\geqslant 0}m_{{s[\varpi]}}(M_{k}(\Gamma[2]))\,t^{k}$
---|---
$s[6]$ | $\frac{1+t^{35}}{(1-t^{4})(1-t^{6})(1-t^{10})(1-t^{12})}$
$s[5,1]$ | $\frac{t^{11}(1+t)}{((1-t^{4})(1-t^{6}))^{2}}$
$s[4,2]$ | $\frac{t^{4}(1+t^{15})}{(1-t^{2})(1-t^{4})^{2}(1-t^{10})}$
$s[4,1^{2}]$ | $\frac{t^{11}(1+t^{4})}{(1-t)(1-t^{4})(1-t^{6})(1-t^{12})}$
$s[3^{2}]$ | $\frac{t^{7}(1+t^{13})}{(1-t^{2})(1-t^{4})(1-t^{6})(1-t^{12})}$
$s[3,2,1]$ | $\frac{t^{8}(1-t^{8})}{(1-t^{2})^{2}(1-t^{5})(1-t^{6})^{2}}$
$s[3,1^{3}]$ | $\frac{t^{6}(1+t^{4}+t^{11}+t^{15})}{(1-t^{2})(1-t^{4})(1-t^{6})(1-t^{12})}$
$s[2^{3}]$ | $\frac{t^{2}(1+t^{23})}{(1-t^{2})(1-t^{4})(1-t^{6})(1-t^{12})}$
$s[2^{2},1^{2}]$ | $\frac{t^{9}}{(1-t^{2})(1-t^{4})^{2}(1-t^{5})}$
$s[2,1^{4}]$ | $\frac{t^{6}(1+t^{11})}{((1-t^{4})(1-t^{6}))^{2}}$
$s[1^{6}]$ | $\frac{t^{5}(1+t^{25})}{(1-t^{4})(1-t^{6})(1-t^{10})(1-t^{12})}$

######  Remark 3.2.

A couple of sanity checks. Firstly,

$\sum_{k\geqslant
0}\sum_{s[\varpi]}\dim(s[\varpi])m_{{s[\varpi]}}(M_{k}(\Gamma[2]))\,t^{k}=\sum_{k\geqslant
0}\dim
M_{k}(\Gamma[2])\,t^{k}=\frac{(1+t^{2})(1+t^{4})(1+t^{5})}{(1-t^{2})^{4}},$

which is in agreement with Theorem 3.1. Secondly, the generating series for
the dimension of spaces of modular forms on $\Gamma_{0}[2]$ is given by (see
[24, Appendix A.1] and the references therein)

$\sum_{k\geqslant 0}\dim
M_{k}(\Gamma_{0}[2])\,t^{k}=\frac{1+t^{19}}{(1-t^{2})(1-t^{4})^{2}(1-t^{6})}.$

We checked that by adding the generating series for the multiplicities of the
irreducible representations $s[6],s[4,2]$ and $s[2^{3}]$ we recover this
formula.

Let us give the first few isotypical decompositions of $M_{k}(\Gamma[2])$, we
put $d=\dim M_{k}(\Gamma[2])$

$\begin{array}[]{c|c|c|c|c|c|c|c|c|c|c|c|c}s[\varpi]&s[6]&s[5,1]&s[4,2]&s[4,1^{2}]&s[3^{2}]&s[3,2,1]&s[3,1^{3}]&s[2^{3}]&s[2^{2},1^{2}]&s[2,1^{4}]&s[1^{6}]&\\\
\dim s[\varpi]&1&5&9&10&5&16&10&5&9&5&1&\\\ \hline\cr\hline\cr
k&&&&&&&&&&&&d\\\ \hline\cr 0&1&0&0&0&0&0&0&0&0&0&0&1\\\
1&0&0&0&0&0&0&0&0&0&0&0&0\\\ 2&0&0&0&0&0&0&0&1&0&0&0&5\\\
3&0&0&0&0&0&0&0&0&0&0&0&0\\\ 4&1&0&1&0&0&0&0&1&0&0&0&15\\\
5&0&0&0&0&0&0&0&0&0&0&1&1\\\ 6&1&0&1&0&0&0&1&2&0&1&0&35\\\
7&0&0&0&0&1&0&0&0&0&0&0&5\\\ 8&1&0&3&0&0&1&1&3&0&0&0&69\\\
9&0&0&0&0&1&0&0&0&1&0&1&15\\\ 10&2&0&3&0&0&2&3&4&0&2&0&121\\\
11&0&1&0&1&2&0&0&0&1&0&1&35\end{array}$

Next, we give the dimension of the various pieces of the space
$M_{k}(\Gamma[2])$ as in (1) and (2) and also their isotypical decomposition.
We distinguish two cases according to the parity of $k$.

###  3.1. Isotypical decomposition of $M_{2k+1}(\Gamma[2])$

We already have seen that $M_{2k+1}(\Gamma[2])=S_{2k+1}(\Gamma[2])$ and from
Theorem 3.1 we deduce $\dim S_{1}(\Gamma[2])=\dim S_{3}(\Gamma[2])=0$ and

$\dim S_{2k+1}(\Gamma[2])=(2k^{3}-9k^{2}+19k-15)/3\quad\text{for}\quad
k\geqslant 2.$

Therefore the generating series for $\dim S_{2k+1}(\Gamma[2])$ is given by

$\sum_{k\geqslant 0}\dim
S_{2k+1}(\Gamma[2])\,t^{2k+1}=\frac{t^{5}(1+t^{2}+t^{4}+t^{6})}{(1-t^{2})^{4}}.$

We have seen that for $k\geqslant 0$ we have
$M_{2k+1}(\Gamma[2])=S_{2k+1}(\Gamma[2])=\chi_{5}\cdot M_{2k-4}(\Gamma[2])$
and since the cusp form $\chi_{5}$ is $\mathfrak{S}_{6}$-anti-invariant (i.e.
it occurs in the alternating representation $s[1^{6}]$) we get

$\dim_{\mathfrak{S}_{6}}M_{2k+1}(\Gamma[2])=\dim_{\mathfrak{S}_{6}}S_{2k+1}(\Gamma[2])=s[1^{6}]\otimes\dim_{\mathfrak{S}_{6}}M_{2k-4}(\Gamma[2]).$

From the generating series of the multiplicities of the irreducible
representations of $\mathfrak{S}_{6}$ previously given, we deduce

$s[\varpi]$ | $\sum_{k\geqslant 0}m_{{s[\varpi]}}(S_{2k+1}(\Gamma[2]))\,t^{2k+1}$
---|---
$s[6]$ | $\frac{t^{35}}{(1-t^{4})(1-t^{6})(1-t^{10})(1-t^{12})}$
$s[5,1]$ | $\frac{t^{11}}{((1-t^{4})(1-t^{6}))^{2}}$
$s[4,2]$ | $\frac{t^{19}}{(1-t^{2})(1-t^{4})^{2}(1-t^{10})}$
$s[4,1^{2}]$ | $\frac{t^{11}(1+t^{4})}{(1-t^{2})(1-t^{4})(1-t^{6})(1-t^{12})}$
$s[3^{2}]$ | $\frac{t^{7}}{(1-t^{2})(1-t^{4})(1-t^{6})(1-t^{12})}$
$s[3,2,1]$ | $\frac{t^{13}(1+t^{2}+t^{4}+t^{6})}{(1-t^{2})(1-t^{6})^{2}(1-t^{10})}$
$s[3,1^{3}]$ | $\frac{t^{17}(1+t^{4})}{(1-t^{2})(1-t^{4})(1-t^{6})(1-t^{12})}$
$s[2^{3}]$ | $\frac{t^{25}}{(1-t^{2})(1-t^{4})(1-t^{6})(1-t^{12})}$
$s[2^{2},1^{2}]$ | $\frac{t^{9}}{(1-t^{2})(1-t^{4})^{2}(1-t^{10})}$
$s[2,1^{4}]$ | $\frac{t^{17}}{((1-t^{4})(1-t^{6}))^{2}}$
$s[1^{6}]$ | $\frac{t^{5}}{(1-t^{4})(1-t^{6})(1-t^{10})(1-t^{12})}$

Let us give the first few isotypical decompositions of $S_{2k+1}(\Gamma[2])$,
we put $d=\dim S_{2k+1}(\Gamma[2])$.

$\begin{array}[]{c|c|c|c|c|c|c|c|c|c|c|c|c}s[\varpi]&s[6]&s[5,1]&s[4,2]&s[4,1^{2}]&s[3^{2}]&s[3,2,1]&s[3,1^{3}]&s[2^{3}]&s[2^{2},1^{2}]&s[2,1^{4}]&s[1^{6}]&\\\
\dim s[\varpi]&1&5&9&10&5&16&10&5&9&5&1&\\\ \hline\cr\hline\cr
2k+1&&&&&&&&&&&&d\\\ \hline\cr 1&0&0&0&0&0&0&0&0&0&0&0&0\\\
3&0&0&0&0&0&0&0&0&0&0&0&0\\\ 5&0&0&0&0&0&0&0&0&0&0&1&1\\\
7&0&0&0&0&1&0&0&0&0&0&0&5\\\ 9&0&0&0&0&1&0&0&0&1&0&1&15\\\
11&0&1&0&1&2&0&0&0&1&0&1&35\\\ 13&0&0&0&1&3&1&0&0&3&0&1&69\\\
15&0&2&0&3&4&2&0&0&3&0&2&121\end{array}$

We further refine these formulas according to the decomposition (2).
Conjecture 6.6 of [4], now proved by Rösner, see [23, Section 5], tells us
that only the Arthur packets (P) and (G) can occur in $S_{2k+1}(\Gamma[2])$.
Hence,

$S_{2k+1}(\Gamma[2])=S^{\textbf{(G)}}_{2k+1}(\Gamma[2])\oplus
S^{\textbf{(P)}}_{2k+1}(\Gamma[2]).$

Moreover this conjecture gives the isotypical decomposition of
$S^{\textbf{(P)}}_{2k+1}(\Gamma[2])$:

$\dim_{\mathfrak{S}_{6}}S^{\textbf{(P)}}_{2k+1}(\Gamma[2])=d^{-}_{2,4k}s[5,1]+d_{4,4k}s[3^{2}]+d^{+}_{2,4k}s[1^{6}]$

where

$d_{N,k}=\dim S_{k}(\Gamma_{0}(N))^{\text{new}}\quad\text{and}\quad
d^{\pm}_{N,k}=\dim S^{\pm}_{k}(\Gamma_{0}(N))^{\text{new}}.$ (4)

Here $\Gamma_{0}(N)$ denotes the Hecke congruence subgroup of level $N$ of
${\mathrm{SL}}(2,\mathbb{Z})$ and $S^{\pm}_{k}(\Gamma_{0}(N))^{\text{new}}$
denotes the space of new cusp forms of weight $k$ on $\Gamma_{0}(N)$ with
eigenvalues $\pm 1$ for the Fricke involution. Note that for $N=1$, we have
$d_{1,k}=\dim S_{k}(\Gamma_{0}(1))^{\text{new}}=\dim
S_{k}({\mathrm{SL}}(2,\mathbb{Z}))$. The dimension of the space
$S_{k}(\Gamma_{0}(N))^{\text{new}}$ is classical, see [27, Chapter 6] and for
example we have:

$\dim S_{4k}(\Gamma_{0}(2))^{\text{new}}=k-1-2\,\lfloor
k/3\rfloor\quad\text{and}\quad\dim S_{4k}(\Gamma_{0}(4))^{\text{new}}=\lfloor
k/3\rfloor\quad\text{for}\quad k\geqslant 1.$ (5)

For $k>2$, the dimensions of $S^{\pm}_{k}(\Gamma_{0}(2))^{\text{new}}$ are
given by (see [20, Theorem 2.2])

$d^{\pm}_{2,k}=\left\\{\begin{tabular}[]{ccc}$(d_{2,k}\pm
1)/2$&\text{if}&$k\equiv 0\mod 8$\\\ $(d_{2,k}\mp 1)/2$&\text{if}&$k\equiv
2\mod 8$\\\ $d_{2,k}/2$&\text{if}&$k\equiv 4,6\mod 8$.\end{tabular}\right.$

So the generating series for the multiplicities of the irreducible
representations $s[5,1]$, $s[3^{2}]$ and $s[1^{6}]$ in
$S^{\textbf{(P)}}_{2k+1}(\Gamma[2])$ for $k\geqslant 0$ are given by

$s[\varpi]$ | $s[5,1]$ | $s[3^{2}]$ | $s[1^{6}]$
---|---|---|---
$\sum_{k\geqslant 0}m_{{s[\varpi]}}(S^{\textbf{(P)}}_{2k+1}(\Gamma[2]))\,t^{2k+1}$ | $\frac{t^{11}}{(1-t^{4})(1-t^{6})}$ | $\frac{t^{7}}{(1-t^{2})(1-t^{6})}$ | $\frac{t^{5}}{(1-t^{4})(1-t^{6})}$

Keeping in mind that $\dim s[5,1]=\dim s[3^{2}]=5$ and $\dim s[1^{6}]=1$, we
get

$\sum_{k\geqslant 0}\dim
S^{\textbf{(P)}}_{2k+1}(\Gamma[2])t^{2k+1}=\frac{t^{5}+5t^{7}+5t^{9}+5t^{11}}{(1-t^{4})(1-t^{6})}.$

From this we deduce the generating series for $\dim
S^{\textbf{(G)}}_{2k+1}(\Gamma[2])$

$\displaystyle\sum_{k\geqslant 0}\dim
S^{\textbf{(G)}}_{2k+1}(\Gamma[2])\,t^{2k+1}=$ $\displaystyle\sum_{k\geqslant
0}\bigl{(}\dim S_{2k+1}(\Gamma[2])-\dim
S^{\textbf{(P)}}_{2k+1}(\Gamma[2])\bigr{)}\,t^{2k+1}=\frac{t^{9}(t^{8}-2t^{6}+10t^{4}+6t^{2}+9)}{(1-t^{2})^{3}(1-t^{6})(1+t^{2})}.$

We also deduce the generating series for the multiplicities of the irreducible
representations in $S^{\textbf{(G)}}_{2k+1}(\Gamma[2])$ for $k\geqslant 0$

  * •

for
$s[\varpi]\in\left\\{s[6],s[4,2],s[4,1^{2}],s[3,2,1],s[3,1^{3}],s[2^{3}],s[2^{2},1^{2}],s[2,1^{4}]\right\\}$
we have

$m_{{s[\varpi]}}(S^{\textbf{(G)}}_{2k+1}(\Gamma[2]))=m_{{s[\varpi]}}(S_{2k+1}(\Gamma[2]))$

  * •

for $s[\varpi]\in\left\\{s[5,1],s[3^{2}],s[1^{6}]\right\\}$ we have

$m_{{s[\varpi]}}(S^{\textbf{(G)}}_{2k+1}(\Gamma[2]))=m_{{s[\varpi]}}(S_{2k+1}(\Gamma[2]))-m_{{s[\varpi]}}(S^{\textbf{(P)}}_{2k+1}(\Gamma[2]))$

so

$s[\varpi]$ | $s[5,1]$ | $s[3^{2}]$ | $s[1^{6}]$
---|---|---|---
$\sum_{k\geqslant 0}m_{{s[\varpi]}}(S^{\textbf{(G)}}_{2k+1}(\Gamma[2]))\,t^{2k+1}$ | $\frac{t^{15}(1+t^{2}-t^{6})}{((1-t^{4})(1-t^{6}))^{2}}$ | $\frac{t^{11}(1+t^{8}-t^{12})}{(1-t^{2})(1-t^{4})(1-t^{6})(1-t^{12})}$ | $\frac{t^{15}(1+t^{2}-t^{12})}{(1-t^{4})(1-t^{6})(1-t^{10})(1-t^{12})}$

The next table gives the first few isotypical decompositions of
$S_{2k+1}(\Gamma[2])$, we indicate the multiplicities of Saito-Kurakawa lifts
in blue and those of the general type in black, we put
${\color[rgb]{0,0,1}d_{P}}=\dim S^{\textbf{(P)}}_{k}(\Gamma[2])$ and
$d_{G}=\dim S^{\textbf{(G)}}_{k}(\Gamma[2])$

$\begin{array}[]{c|c|c|c|c|c|c|c|c|c|c|c|c}s[\varpi]&s[6]&s[5,1]&s[4,2]&s[4,1^{2}]&s[3^{2}]&s[3,2,1]&s[3,1^{3}]&s[2^{3}]&s[2^{2},1^{2}]&s[2,1^{4}]&s[1^{6}]&\\\
\dim s[\varpi]&1&5&9&10&5&16&10&5&9&5&1&\\\ \hline\cr\hline\cr
2k+1&&&&&&&&&&&&d_{G}+{\color[rgb]{0,0,1}d_{P}}\\\ \hline\cr
1&0&0&0&0&0&0&0&0&0&0&0&0+{\color[rgb]{0,0,1}0}\\\
3&0&0&0&0&0&0&0&0&0&0&0&0+{\color[rgb]{0,0,1}0}\\\
5&0&0&0&0&0&0&0&0&0&0&{\color[rgb]{0,0,1}1}&0+{\color[rgb]{0,0,1}1}\\\
7&0&0&0&0&{\color[rgb]{0,0,1}1}&0&0&0&0&0&0&0+{\color[rgb]{0,0,1}5}\\\
9&0&0&0&0&{\color[rgb]{0,0,1}1}&0&0&0&1&0&{\color[rgb]{0,0,1}1}&9+{\color[rgb]{0,0,1}6}\\\
11&0&{\color[rgb]{0,0,1}1}&0&1&1+{\color[rgb]{0,0,1}1}&0&0&0&1&0&{\color[rgb]{0,0,1}1}&24+{\color[rgb]{0,0,1}11}\\\
13&0&0&0&1&1+{\color[rgb]{0,0,1}2}&1&0&0&3&0&{\color[rgb]{0,0,1}1}&58+{\color[rgb]{0,0,1}11}\\\
15&0&1+{\color[rgb]{0,0,1}1}&0&3&2+{\color[rgb]{0,0,1}2}&2&0&0&3&0&1+{\color[rgb]{0,0,1}1}&105+{\color[rgb]{0,0,1}16}\end{array}$

###  3.2. Isotypical decomposition of $M_{2k}(\Gamma[2])$

We start by giving the generating series for the multiplicities of the
irreducible representations of $\mathfrak{S}_{6}$ in $M_{2k}(\Gamma[2])$.

$s[\varpi]$ | $\sum_{k\geqslant 0}m_{{s[\varpi]}}(M_{2k}(\Gamma[2]))\,t^{2k}$
---|---
$s[6]$ | $\frac{1}{(1-t^{4})(1-t^{6})(1-t^{10})(1-t^{12})}$
$s[5,1]$ | $\frac{t^{12}}{((1-t^{4})(1-t^{6}))^{2}}$
$s[4,2]$ | $\frac{t^{4}}{(1-t^{2})(1-t^{4})^{2}(1-t^{10})}$
$s[4,1^{2}]$ | $\frac{t^{12}(1+t^{4})}{(1-t^{2})(1-t^{4})(1-t^{6})(1-t^{12})}$
$s[3^{2}]$ | $\frac{t^{20}}{(1-t^{2})(1-t^{4})(1-t^{6})(1-t^{12})}$
$s[3,2,1]$ | $\frac{t^{8}(1+t^{2}+t^{4}+t^{8})}{(1-t^{2})(1-t^{6})^{2}(1-t^{10})}$
$s[3,1^{3}]$ | $\frac{t^{6}(1+t^{4})}{(1-t^{2})(1-t^{4})(1-t^{6})(1-t^{12})}$
$s[2^{3}]$ | $\frac{t^{2}}{(1-t^{2})(1-t^{4})(1-t^{6})(1-t^{12})}$
$s[2^{2},1^{2}]$ | $\frac{t^{14}}{(1-t^{2})(1-t^{4})^{2}(1-t^{10})}$
$s[2,1^{4}]$ | $\frac{t^{6}}{((1-t^{4})(1-t^{6}))^{2}}$
$s[1^{6}]$ | $\frac{t^{30}}{(1-t^{4})(1-t^{6})(1-t^{10})(1-t^{12})}$

######  Remark 3.3.

As a sanity check, we verify

$\sum_{k\geqslant
0}\sum_{s[\varpi]}\dim(s[\varpi])m_{{s[\varpi]}}(M_{2k}(\Gamma[2]))\,t^{2k}=\frac{(1+t^{2})(1+t^{4})}{(1-t^{2})^{4}}=\sum_{k\geqslant
0}\dim M_{2k}(\Gamma[2])\,t^{2k}$

in agreement with Theorem 3.1.

Let us give the first few isotypical decompositions of $M_{2k}(\Gamma[2])$, we
put $d=\dim M_{2k}(\Gamma[2])$

$\begin{array}[]{c|c|c|c|c|c|c|c|c|c|c|c|c}s[\varpi]&s[6]&s[5,1]&s[4,2]&s[4,1^{2}]&s[3^{2}]&s[3,2,1]&s[3,1^{3}]&s[2^{3}]&s[2^{2},1^{2}]&s[2,1^{4}]&s[1^{6}]&\\\
\dim s[\varpi]&1&5&9&10&5&16&10&5&9&5&1&\\\ \hline\cr\hline\cr
k&&&&&&&&&&&&d\\\ \hline\cr 0&1&0&0&0&0&0&0&0&0&0&0&1\\\
2&0&0&0&0&0&0&0&1&0&0&0&5\\\ 4&1&0&1&0&0&0&0&1&0&0&0&15\\\
6&1&0&1&0&0&0&1&2&0&1&0&35\\\ 8&1&0&3&0&0&1&1&3&0&0&0&69\\\
10&2&0&3&0&0&2&3&4&0&2&0&121\\\ 12&3&1&6&1&0&3&4&5&0&2&0&195\\\
14&2&0&7&1&0&6&6&8&1&3&0&295\\\ 16&4&2&11&3&0&8&8&9&1&4&0&425\end{array}$

From (1) we have the following decomposition of the space $M_{2k}(\Gamma[2])$:

$M_{2k}(\Gamma[2])=E_{2k}(\Gamma[2])\oplus
S_{2k}(\Gamma[2])\quad\text{for}\quad k\geqslant 0$

so we are going to give the isotypical decomposition of the Eisenstein and the
cuspidal parts according to (2).

####  3.2.1. Isotypical decomposition of $E_{2k}(\Gamma[2])$

From (2), we have

$E_{2k}(\Gamma[2])=E^{\textbf{(F)}}_{2k}(\Gamma[2])\oplus
E^{\textbf{(Q)}}_{2k}(\Gamma[2])\quad\text{for}\quad k\geqslant 0.$

Recall that $E^{\textbf{(F)}}_{2k}(\Gamma[2])$ corresponds to Siegel-
Eisenstein series while $E^{\textbf{(Q)}}_{2k}(\Gamma[2])$ to Klingen-
Eisenstein series. From [12, Section 13], we deduce that

$\dim_{\mathfrak{S}_{6}}E^{\textbf{(F)}}_{0}(\Gamma[2])=s[6],\;\dim_{\mathfrak{S}_{6}}E^{\textbf{(F)}}_{2}(\Gamma[2])=s[2^{3}]\text{
and
}\dim_{\mathfrak{S}_{6}}E^{\textbf{(F)}}_{2k}(\Gamma[2])=s[6]+s[4,2]+s[2^{3}]\text{\,
for\, }k\geqslant 2.$

By construction, Klingen-Eisenstein series of weight $2k$ for $k>2$ (to ensure
convergence) on $\Gamma[2]$ come from cusp forms of weight $2k$ on
$\Gamma(2)$. Since
${\mathrm{SL}}(2,\mathbb{Z})/\Gamma(2)\cong\mathfrak{S}_{3}$, the space
$S_{2k}(\Gamma(2))$ can be decomposed into irreducible representations of
$\mathfrak{S}_{3}$ and this is given by

$\dim_{\mathfrak{S}_{3}}S_{2k}(\Gamma(2))=d_{1,2k}s[3]+(d_{1,2k}+d_{2,2k})s[2,1]+d_{4,2k}s[1^{3}],$
(6)

where the integers $d_{N,k}$ are defined as in (4). Note that the previous
formula corrects [21, Proposition 6.1]. Recall that the dimension of the space
$S_{2k}(\Gamma(2))\cong S_{2k}(\Gamma_{0}(4))$ is $k-2$ for $k\geqslant 3$ and
$0$ otherwise.

From the previous table, Theorem 3.1 and the isotypical decomposition of
$E^{\textbf{(F)}}_{2k}(\Gamma[2])$, we deduce that
$\dim_{\mathfrak{S}_{6}}E^{\textbf{(Q)}}_{0}(\Gamma[2])=\dim_{\mathfrak{S}_{6}}E^{\textbf{(Q)}}_{2}(\Gamma[2])=0.$
For $k\geqslant 2$, Proposition 13.1 of [12] gives

$\dim_{\mathfrak{S}_{6}}E^{\textbf{(Q)}}_{2k}(\Gamma[2])=\text{Ind}_{H}^{\mathfrak{S}_{6}}\left(\dim_{\mathfrak{S}_{3}}S_{2k}(\Gamma(2))\right)$

where $H$ denotes the stabiliser in $\mathfrak{S}_{6}$ of one of the
$1$-dimensional boundary components of the Satake compactification of
$\Gamma[2]\backslash\mathfrak{H}_{2}$. Note that $H$ is of order 48 and recall
(for more details see [12, Section 2]) that we have

$s[\varpi]$ | $s[3]$ | $s[2,1]$ | $s[1^{3}]$
---|---|---|---
${\rm Ind}_{H}^{\mathfrak{S}_{6}}\left(s[\varpi]\right)$ | $s[6]\oplus s[5,1]\oplus s[4,2]$ | $s[4,2]\oplus s[3,2,1]\oplus s[2^{3}]$ | $s[3,1^{3}]\oplus s[2,1^{4}]$

Therefore, for $k\geqslant 2$, the isotypical decomposition of the space
$E^{\textbf{(Q)}}_{2k}(\Gamma[2])$ is as follows

$\displaystyle\dim_{\mathfrak{S}_{6}}E^{\textbf{(Q)}}_{2k}(\Gamma[2])=$
$\displaystyle
d_{1,2k}(s[6]+s[5,1])+(2d_{1,2k}+d_{2,2k})s[4,2]+(d_{1,2k}+d_{2,2k})(s[3,2,1]+s[2^{3}])$
$\displaystyle+d_{4,2k}(s[3,1^{3}]+s[2,1^{4}]).$

Putting this together we get the generating series for the multiplicities of
the irreducible representations of $\mathfrak{S}_{6}$ in
$E^{\textbf{(F)}}_{2k}(\Gamma[2])$ and $E^{\textbf{(Q)}}_{2k}(\Gamma[2])$

$s[\varpi]$ | $\sum_{k\geqslant 0}m_{{s[\varpi]}}(E^{\textbf{(F)}}_{2k}(\Gamma[2]))\,t^{2k}$ | $\sum_{k\geqslant 0}m_{{s[\varpi]}}(E^{\textbf{(Q)}}_{2k}(\Gamma[2]))\,t^{2k}$
---|---|---
$s[6]$ | $\frac{1-t^{2}+t^{4}}{1-t^{2}}$ | $\frac{t^{12}}{(1-t^{4})(1-t^{6})}$
$s[5,1]$ | $0$ | $\frac{t^{12}}{(1-t^{4})(1-t^{6})}$
$s[4,2]$ | $\frac{t^{4}}{1-t^{2}}$ | $\frac{t^{8}}{(1-t^{2})(1-t^{4})}$
$s[3,2,1]$ | $0$ | $\frac{t^{8}}{(1-t^{2})(1-t^{6})}$
$s[3,1^{3}]$ | $0$ | $\frac{t^{6}}{(1-t^{4})(1-t^{6})}$
$s[2^{3}]$ | $\frac{t^{2}}{1-t^{2}}$ | $\frac{t^{8}}{(1-t^{2})(1-t^{6})}$
$s[2,1^{4}]$ | $0$ | $\frac{t^{6}}{(1-t^{4})(1-t^{6})}$

and $0$ for $s[4,1^{2}],s[3^{2}],s[2^{2},1^{2}]$ and $s[1^{6}]$.

######  Remark 3.4.

The isotypical decomposition of the space $S_{2k}(\Gamma(2))$ in [12,
Proposition 13.1] was written as

$\dim_{\mathfrak{S}_{3}}S_{2k}(\Gamma(2))=\text{{Sym}}^{k}(s[2,1])-\left\\{\begin{tabular}[]{ccc}$s[2,1]$&\text{if}&$k=1$\\\
$s[3]+s[2,1]$&\text{if}&$k\geqslant 2$.\end{tabular}\right.$

This directly gives $\dim S_{4k}(\Gamma(2))=2(k-1)$ for $k\geqslant 1$, this
can also be checked by using (5), (6) and $\dim
S_{4k}({\mathrm{SL}}(2,\mathbb{Z}))=\lfloor k/3\rfloor$. As a sanity check, we
verify

$\displaystyle\sum_{k\geqslant
0}\sum_{s[\varpi]}\dim(s[\varpi])m_{{s[\varpi]}}(E^{\textbf{(Q)}}_{2k}(\Gamma[2]))t^{2k}$
$\displaystyle=15\frac{t^{6}}{(1-t^{2})^{2}}=15\sum_{k\geqslant 0}\dim
S_{2k}(\Gamma(2))t^{2k}$

in agreement with $E^{\textbf{(Q)}}_{2k}(\Gamma[2])\cong
S_{2k}(\Gamma(2))^{\oplus 15}$.

The next table gives the first few isotypical decompositions of
$E_{2k}(\Gamma[2])$ where we indicate the multiplicities of the Siegel-
Eisenstein part in blue and those of the Klingen-Eisenstein part in black, we
put ${\color[rgb]{0,0,1}d_{F}}=\dim E^{\textbf{(F)}}_{k}(\Gamma[2])$ and
$d_{Q}=\dim E^{\textbf{(Q)}}_{k}(\Gamma[2])$

$\begin{array}[]{c|c|c|c|c|c|c|c|c|c|c|c|c}s[\varpi]&s[6]&s[5,1]&s[4,2]&s[4,1^{2}]&s[3^{2}]&s[3,2,1]&s[3,1^{3}]&s[2^{3}]&s[2^{2},1^{2}]&s[2,1^{4}]&s[1^{6}]&\\\
\dim s[\varpi]&1&5&9&10&5&16&10&5&9&5&1&\\\ \hline\cr\hline\cr
k&&&&&&&&&&&&d={\color[rgb]{0,0,1}d_{F}}+d_{Q}\\\ \hline\cr
0&{\color[rgb]{0,0,1}1}&0&0&0&0&0&0&0&0&0&0&{\color[rgb]{0,0,1}1}+0\\\
2&0&0&0&0&0&0&0&{\color[rgb]{0,0,1}1}&0&0&0&{\color[rgb]{0,0,1}5}+0\\\
4&{\color[rgb]{0,0,1}1}&0&{\color[rgb]{0,0,1}1}&0&0&0&0&{\color[rgb]{0,0,1}1}&0&0&0&{\color[rgb]{0,0,1}15}+0\\\
6&{\color[rgb]{0,0,1}1}&0&{\color[rgb]{0,0,1}1}&0&0&0&1&{\color[rgb]{0,0,1}1}&0&1&0&{\color[rgb]{0,0,1}15}+15\\\
8&{\color[rgb]{0,0,1}1}&0&{\color[rgb]{0,0,1}1}+1&0&0&1&0&{\color[rgb]{0,0,1}1}+1&0&0&0&{\color[rgb]{0,0,1}15}+30\\\
10&{\color[rgb]{0,0,1}1}&0&{\color[rgb]{0,0,1}1}+1&0&0&1&1&{\color[rgb]{0,0,1}1}+1&0&1&0&{\color[rgb]{0,0,1}15}+45\\\
12&{\color[rgb]{0,0,1}1}+1&1&{\color[rgb]{0,0,1}1}+2&0&0&1&1&{\color[rgb]{0,0,1}1}+1&0&1&0&{\color[rgb]{0,0,1}15}+60\\\
14&{\color[rgb]{0,0,1}1}&0&{\color[rgb]{0,0,1}1}+2&0&0&2&1&{\color[rgb]{0,0,1}1}+2&0&1&0&{\color[rgb]{0,0,1}15}+75\\\
\end{array}$

####  3.2.2. Isotypical decomposition of $S_{2k}(\Gamma[2])$

For $k\geqslant 2$, we know, see [28, pp. 882-883], that

$\dim S_{2k}(\Gamma[2])=\dim
M_{2k}(\Gamma[2])-15(k-2)-15=(k-2)(2k^{2}+7k-24)/3.$ (7)

We also know $S_{0}(\Gamma[2])=S_{2}(\Gamma[2])=\\{0\\}$. The generating
series for the dimension of the spaces $S_{2k}(\Gamma[2])$ is therefore given
by

$\sum_{k\geqslant 0}\dim
S_{2k}(\Gamma[2])\,t^{2k}=\frac{t^{6}(5+4t^{2}-5t^{4})}{(1-t^{2})^{4}}.$

By definition of cusp forms, for $k\geqslant 0$ we have

$m_{{s[\varpi]}}(S_{2k}(\Gamma[2]))=m_{{s[\varpi]}}(M_{2k}(\Gamma[2]))-\Big{(}m_{{s[\varpi]}}(E^{\textbf{(F)}}_{2k}(\Gamma[2]))+m_{{s[\varpi]}}(E^{\textbf{(Q)}}_{2k}(\Gamma[2]))\Big{)}.$

So the generating series for the multiplicities of the irreducible
representations of $\mathfrak{S}_{6}$ in $S_{2k}(\Gamma[2])$ are given by

$s[\varpi]$ | $\sum_{k\geqslant 0}m_{{s[\varpi]}}(S_{2k}(\Gamma[2]))\,t^{2k}$
---|---
$s[6]$ | $\frac{t^{10}(1+t^{2}-t^{12})}{(1-t^{4})(1-t^{6})(1-t^{10})(1-t^{12})}$
$s[5,1]$ | $\frac{t^{16}(1+t^{2}-t^{6})}{((1-t^{4})(1-t^{6}))^{2}}$
$s[4,2]$ | $\frac{t^{8}(1+t^{6}-t^{10})}{(1-t^{2})(1-t^{4})^{2}(1-t^{10})}$
$s[4,1^{2}]$ | $\frac{t^{12}(1+t^{4})}{(1-t^{2})(1-t^{4})(1-t^{6})(1-t^{12})}$
$s[3^{2}]$ | $\frac{t^{20}}{(1-t^{2})(1-t^{4})(1-t^{6})(1-t^{12})}$
$s[3,2,1]$ | $\frac{t^{10}(1+t^{2}+2t^{4}+t^{8}-t^{14})}{(1-t^{2})(1-t^{6})^{2}(1-t^{10})}$
$s[3,1^{3}]$ | $\frac{t^{8}(1+t^{2}+t^{10}-t^{12})}{(1-t^{2})(1-t^{4})(1-t^{6})(1-t^{12})}$
$s[2^{3}]$ | $\frac{t^{6}(1+t^{8}-t^{12})}{(1-t^{2})(1-t^{4})(1-t^{6})(1-t^{12})}$
$s[2^{2},1^{2}]$ | $\frac{t^{14}}{(1-t^{2})(1-t^{4})^{2}(1-t^{10})}$
$s[2,1^{4}]$ | $\frac{t^{10}(1+t^{2}-t^{6})}{((1-t^{4})(1-t^{6}))^{2}}$
$s[1^{6}]$ | $\frac{t^{30}}{(1-t^{4})(1-t^{6})(1-t^{10})(1-t^{12})}$

Again, Conjecture 6.6 of [4] now proved by Rösner (see [23, Section 5]) tells
us that only the Arthur packets (P) and (G) can occur in $S_{2k}(\Gamma[2])$
and so

$S_{2k}(\Gamma[2])=S^{\textbf{(G)}}_{2k}(\Gamma[2])\oplus
S^{\textbf{(P)}}_{2k}(\Gamma[2]).$

Moreover Conjecture 6.6 of [4] gives the isotypical decomposition of
$S^{\textbf{(P)}}_{2k}(\Gamma[2])$:

$\displaystyle\dim_{\mathfrak{S}_{6}}S^{\textbf{(P)}}_{2k}(\Gamma[2])=$
$\displaystyle
d_{1,4k-2}\,s[6]+(d_{1,4k-2}+d^{+}_{2,4k-2})\,s[4,2]+(d_{2,4k-2}+d^{-}_{2,4k-2})\,s[2^{3}]$

where the integers $d_{N,k}$ and $d^{\pm}_{N,k}$ are defined as in (4). The
generating series for the multiplicity of the irreducible representations
$s[6]$, $s[4,2]$ and $s[2^{3}]$ in $S^{\textbf{(P)}}_{2k}(\Gamma[2])$ for
$k\geqslant 0$ are therefore given by

$s[\varpi]$ | $s[6]$ | $s[4,2]$ | $s[2^{3}]$
---|---|---|---
$\sum_{k\geqslant 0}m_{{s[\varpi]}}(S^{\textbf{(P)}}_{2k}(\Gamma[2]))\,t^{2k}$ | $\frac{t^{10}}{(1-t^{2})(1-t^{6})}$ | $\frac{t^{8}}{(1-t^{2})(1-t^{4})}$ | $\frac{t^{6}}{(1-t^{2})(1-t^{4})}$

Keeping in mind that $\dim s[6]=1$, $\dim s[4,2]=9$ and $\dim s[2^{3}]=5$, we
deduce

$\sum_{k\geqslant 0}\dim
S^{\textbf{(P)}}_{2k}(\Gamma[2])t^{2k}=\frac{t^{6}(5+14t^{2}+15t^{4}+10t^{6})}{(1-t^{4})(1-t^{6})}.$

From this we get the generating series for $\dim
S^{\textbf{(G)}}_{2k}(\Gamma[2])$:

$\sum_{k\geqslant 0}\dim
S^{\textbf{(G)}}_{2k}(\Gamma[2])t^{2k}=\sum_{k\geqslant 0}\bigl{(}\dim
S_{2k}(\Gamma[2])-\dim S^{\textbf{(P)}}_{2k}(\Gamma[2])\bigr{)}t^{2k}\\\
=\frac{t^{8}(10+21t^{2}+9t^{4}-t^{6}-15t^{8})}{(1-t^{2})^{2}(1-t^{4})(1-t^{6})}$

and also the generating series for the multiplicity of the irreducible
representations in $S^{\textbf{(G)}}_{2k}(\Gamma[2])$ for $k\geqslant 0$

  * •

for
$s[\varpi]\in\left\\{s[5,1],s[4,1^{2}],s[3^{2}],s[3,2,1],s[3,1^{3}],s[2^{2},1^{2}],s[2,1^{4}],s[1^{6}]\right\\}$
we have

$m_{{s[\varpi]}}(S^{\textbf{(G)}}_{2k}(\Gamma[2]))=m_{{s[\varpi]}}(S_{2k}(\Gamma[2]))$

  * •

for $s[\varpi]\in\left\\{s[6],s[4,2],s[2^{3}]\right\\}$ we have

$m_{{s[\varpi]}}(S^{\textbf{(G)}}_{2k}(\Gamma[2]))=m_{{s[\varpi]}}(S_{2k}(\Gamma[2]))-m_{{s[\varpi]}}(S^{\textbf{(P)}}_{2k}(\Gamma[2]))$

so

$s[\varpi]$ | $s[6]$ | $s[4,2]$ | $s[2^{3}]$
---|---|---|---
$\sum_{k\geqslant 0}m_{{s[\varpi]}}(S^{\textbf{(G)}}_{2k}(\Gamma[2]))\,t^{2k}$ | $\frac{t^{20}(1+t^{2}+t^{4}-t^{12}-t^{14})}{(1-t^{4})(1-t^{6})(1-t^{10})(1-t^{12})}$ | $\frac{t^{12}(1+t^{2}-t^{10})}{(1-t^{2})(1-t^{4})^{2}(1-t^{10})}$ | $\frac{t^{12}(1+t^{2}-t^{12})}{(1-t^{2})(1-t^{4})(1-t^{6})(1-t^{12})}$

##  4\. Euler characteristics of local systems

Let $\mathcal{A}_{2}[2]$ be the moduli space of principally polarized abelian
surfaces equipped with a full level two structure. This is a smooth Deligne-
Mumford stack defined over ${\rm Spec}(\mathbb{Z}[1/2])$. The space
$\mathcal{A}_{2}[2]$ comes equipped with a natural action of the symmetric
group $\mathfrak{S}_{6}\cong{\rm GSp}(4,\mathbb{Z}/2)$. Let
$\pi:\mathcal{X}\to\mathcal{A}_{2}[2]$ denote the universal object and define
the local system $\mathbb{V}=R^{1}\pi_{*}\mathbb{C}$ on
$(\mathcal{A}_{2}[2])_{\mathbb{C}}$. To each pair of integers $(l,m)$, with
$l\geq m\geq 0$, we get a local system $\mathbb{V}_{l,m}$ from the
corresponding irreducible representation of $\mathrm{GSp}(4)$.

The moduli space $\mathcal{A}_{2}[2]$ can be identified with the union of the
moduli space of tuples $(C,r_{1},\ldots,r_{6})$ where $C$ is either a genus
$2$ curve or an unordered pair of elliptic curves (note that curves are
assumed to be projective, irreducible and smooth) intersecting in the point at
infinity, and where $(r_{1},\ldots,r_{6})$ is a $6$-tuple of marked Weierstraß
points (distinct from infinity, in the elliptic curve case). Denote the strata
respectively by $\mathcal{M}_{2}[2]$ and $\mathcal{A}_{1,1}[2]$. With this
identification, the action of $\mathfrak{S}_{6}$ is by permutation of the
marked Weierstraß points. For more details about the above cf. [4].

Let $\psi:(\mathcal{A}_{2}[2])_{\mathbb{C}}\to A_{2}[2]$ denote the coarse
moduli space, respectively $M_{2}[2]$ and $A_{1,1}[2]$, and put
$V_{l,m}=\psi_{*}\mathbb{V}_{l,m}$. There is an induced action of
$\mathfrak{S}_{6}$ on the compactly supported Betti cohomology groups
$H^{i}_{c}$ of these spaces with coefficients in $V_{l,m}$. For any partition
$\varpi=[6^{\varpi_{6}},\ldots,1^{\varpi_{1}}]$ of $6$, put
$h^{i}_{c,\varpi}=m_{s[\varpi]}(H^{i}_{c})$. We will now identify the
representation ring $\mathbb{Z}[\mathfrak{S}_{6}]$ of $\mathfrak{S}_{6}$ with
the ring of symmetric polynomials. With this interpretation, for a partition
$\varpi$ of $6$, $s[\varpi]$ equals the corresponding Schur polynomial. Let
also $p_{i}$ denote the $i$th power sum polynomial and put
$p_{\varpi}=p_{1}^{\varpi_{1}}\cdots p_{6}^{\varpi_{6}}$.

Moreover, for any $\lambda=(l,m)$, let $s_{<\lambda>}$ denote the symplectic
Schur polynomial in four variables associated to $\lambda$, see [14, Appendix
A].

###  4.1. Formulas for the Euler characteristics

The aim of this section is to give a formula, for any $\lambda=(l,m)$, of the
$\mathfrak{S}_{6}$-equivariant Euler characteristic,

$E_{c}(A_{2}[2],V_{\lambda})=\sum_{\varpi\vdash
6}E_{c,\varpi}(A_{2}[2],V_{\lambda})s[\varpi]\in\mathbb{Z}[\mathfrak{S}_{6}],$

where

$E_{c,\varpi}(A_{2}[2],V_{\lambda})=\sum^{4}_{i=0}(-1)^{i}h^{i}_{c,\varpi}(A_{2}[2],V_{\lambda})\in\mathbb{Z}.$

Stratify the spaces $X_{1}=M_{2}[2]$ and $X_{2}=A_{1,1}[2]$ (or equivalently
$M_{2}$ and $A_{1,1}$, the corresponding coarse moduli spaces without a level
two structure), into strata $\Sigma_{i}(G)$, for $G$ a finite group,
consisting of the curves corresponding to points of $X_{i}$ whose automorphism
group equals $G$. Let $E_{c}(\Sigma_{i}(G))$ denote the Euler characteristic
of $\Sigma_{i}(G)$. Say that $g\in G$ has eigenvalues $\xi_{1}(g)$,
$\xi_{2}(g)$, $\xi_{3}(g)$ and $\xi_{4}(g)$ when acting on
$H^{1}(C,\mathbb{C})$ of a curve $C\in\Sigma_{i}(G)$. Say furthermore that the
induced action of $g\in G$ on the six Weierstraß points of a curve
$C\in\Sigma_{i}(G)$ has $\mu_{j}$ cycles of length $j$ for $j=1,\ldots,6$,
giving a partition $\mu(g,G,i)$. Note that this data will be constant on the
strata, i.e. independent of the choice of $C\in\Sigma_{i}(G)$. On a strata
$\Sigma_{i}(G)$ the Euler characteristic
$E_{c}(\Sigma_{i}(G),V_{\lambda})=E_{c}(\Sigma_{i}(G))\cdot\dim
V_{\lambda}^{G}$ and hence

$E_{c}(A_{2},V_{\lambda})=\sum_{i=1}^{2}\sum_{G}\frac{E_{c}\bigl{(}\Sigma_{i}(G)\bigr{)}}{|G|}\sum_{g\in
G}s_{<\lambda>}\bigl{(}\xi_{1}(g),\xi_{2}(g),\xi_{3}(g),\xi_{4}(g)\bigr{)}\in\mathbb{Z}.$
(8)

This method was used in [15] to find a formula for $E_{c}(M_{2},V_{\lambda})$
for any $\lambda$. Adding the level two structure we need to take the action
of $\mathfrak{S}_{6}$ on the Weierstraß points into account and one finds
that,

$E_{c}(A_{2}[2],V_{\lambda})=\sum_{i=1}^{2}\sum_{G}\frac{E_{c}\bigl{(}\Sigma_{i}(G)\bigr{)}}{|G|}\sum_{g\in
G}s_{<\lambda>}\bigl{(}\xi_{1}(g),\xi_{2}(g),\xi_{3}(g),\xi_{4}(g)\bigr{)}p_{\mu(g,G,i)}\in\mathbb{Z}[\mathfrak{S}_{6}].$
(9)

This formula can be compared to the one in [7, Section 9].

In the two following sections, we will describe how to find the necessary
information to compute (9) for any $\lambda$.

###  4.2. Smooth curves of genus two

The stratification by automorphism group $G$ for $M_{2}$ was found by Bolza
[9], see below. We follow the description in [15, Section 4].

Curves $C$ of genus $2$ are described by equations $C_{f}:y^{2}-f(x)=0$, where
$f$ square-free polynomial of degree $5$ or $6$. The automorphism group
$G_{f}$ of a curve $C_{f}$ is equal to the subgroup of
$\mathrm{SL}(2,\mathbb{C})\times\mathbb{C}^{\times}$ consisting of elements

$(\gamma,u)=\Bigl{(}\Bigl{(}\begin{array}[]{cc}a&b\\\
c&d\end{array}\Bigr{)},u\Bigr{)}\in\mathrm{SL}(2,\mathbb{C})\times\mathbb{C}^{\times}\quad\text{such
that}\quad f(x)=(\gamma,u)\cdot
f(x)=\frac{(cx+d)^{6}}{u^{2}}\,f\Bigl{(}\frac{ax+b}{cx+d}\Bigr{)}$

quotiented by the subgroup generated by the element
$(-\mathrm{id},-1)\in\mathrm{SL}(2,\mathbb{C})\times\mathbb{C}^{\times}$.
These groups will be given as pairs $(\Gamma_{f},\rho_{f})$, where
$\Gamma_{f}$ is a subgroup of $\mathrm{SL}(2,\mathbb{C})$ that preserves the
set of roots of $f$ and $\rho_{f}$ is a character of $\Gamma_{f}$ such that
$G_{f}\cong\Gamma_{f}(\rho_{f})/<(-\mathrm{id},-1)>$ where

$\Gamma_{f}(\rho_{f})=\\{(\gamma,u)\in\mathrm{SL}(2,\mathbb{C})\times\mathbb{C}^{\times}:u^{2}=\rho_{f}(\gamma)\\}.$

There is an isomorphism $H^{1}(C_{f},\mathbb{C})\cong
H^{0}(C_{f},\Omega)\oplus H^{0}(C_{f},\Omega)^{\vee}$ and
$H^{0}(C_{f},\Omega)$ has a basis consisting of the differentials
$\omega_{0}=dx/y$, $\omega_{1}=xdx/y$. The action of
$(\gamma,u)\in\Gamma_{f}(\rho_{f})$ on the basis $(\omega_{0},\omega_{1})$
equals

$(\gamma,u)(\omega_{0},\omega_{1})=(u^{-1}(c\omega_{1}+d\omega_{0}),u^{-1}(a\omega_{1}+b\omega_{0})),$

see [15, Proposition 2]. This tells us that if $\lambda_{\gamma}$ is an
eigenvalues of $\gamma\in\Gamma_{f}$ then $\xi_{1}=\lambda_{\gamma}u^{-1}$,
$\xi_{2}=\lambda_{\gamma}^{-1}u^{-1}$, $\xi_{3}=\lambda_{\gamma}^{-1}u$ and
$\xi_{4}=\lambda_{\gamma}u$, are the eigenvalues of $(\gamma,u)\in G_{f}$
acting on $H^{1}(C_{f},\mathbb{C})$. Finally, we need to determine the action
of every $\gamma\in\Gamma_{f}$ on the roots of $f$, together with the point at
infinity in the case that the degree of $f$ equals five. We will choose an
ordering of the roots of $f$ (and possibly infinity) and denote the induced
permutation by $\sigma_{\gamma}(f)$.

There are seven strata for $M_{2}[2]$ corresponding to the different
automorphism groups $(\Gamma,\rho)$: $(C_{2},\mathrm{id})$,
$(C_{4},\chi^{2})$, $(Q_{8},\chi_{0})$, $(Q_{12},\chi_{0})$, $(O,\chi)$,
$(Q_{24},\chi_{+})$ and $(C_{10},\chi^{6})$. Here $C_{n}$ denotes the cyclic
group with $n$ elements, $Q_{4n}$ the quaternionic group with $4n$ elements
and $O$ is the binary octahedral group with $48$ elements. The characters are
defined as in [15, pp. 124–125]. The groups
$\Gamma\subset\mathrm{SL}(2,\mathbb{C})$ can be generated by one element
$S\in\mathrm{SL}(2,\mathbb{C})$ in the abelian case, and two elements $S$ and
$U=\left(\begin{smallmatrix}0&1\\\ -1&0\end{smallmatrix}\right)$ in the non-
abelian case. Put $\epsilon_{n}=e^{2\pi i/n}$. For further descriptions of
these groups and characters, together with the computation of the Euler
characteristics of the different strata, we refer to [15, Section 2]. The
information in the following table can be gotten from straightforward
computations.

$(\Gamma,\rho)$ $f\in\Sigma(G)$ $E_{c}$ $S$ $\rho(S)$ $\sigma_{S}$ $\rho(U)$
$\sigma_{U}$ $(C_{2},\mathrm{id})$ $-1$ $\mathrm{diag}(-1,-1)$ $1$
$\mathrm{id}$ $(C_{4},\chi^{2})$ $x^{6}+\alpha x^{4}+\beta x^{2}+1$ $3$
$\mathrm{diag}(\epsilon_{4},\epsilon^{-1}_{4})$ $-1$ $(12)(34)(56)$
$(Q_{8},\chi_{0})$ $x(x^{4}+\alpha x^{2}+1)$ $-2$
$\mathrm{diag}(\epsilon_{4},\epsilon^{-1}_{4})$ $1$ $(23)(45)$ $-1$
$(16)(24)(35)$ $(Q_{12},\chi_{0})$ $x^{6}+\alpha x^{3}-1$ $-2$
$\mathrm{diag}(\epsilon_{6},\epsilon^{-1}_{6})$ $1$ $(123)(456)$ $-1$
$(14)(25)(36)$ $(O,\chi)$ $x(x^{4}+1)$ $1$
$\frac{-1}{\sqrt{2}}\Bigl{(}\begin{array}[]{cc}1&\epsilon_{8}\\\
\epsilon_{8}^{3}&1\end{array}\Bigr{)}$ $\epsilon_{8}^{3}$ $(1264)$ $-1$
$(16)(23)(45)$ $(Q_{24},\chi_{+})$ $x^{6}-1$ $1$
$\mathrm{diag}(\epsilon_{12},\epsilon^{-1}_{12})$ $-1$ $(123456)$ $1$
$(16)(25)(34)$ $(C_{10},\chi^{6})$ $x(x^{5}-1)$ $1$
$\mathrm{diag}(\epsilon_{10},\epsilon^{-1}_{10})$ $\epsilon_{10}^{6}$
$(23456)$

The table above provides sufficient information to compute the contribution of
$\Sigma_{1}(G)$ to (9), for all abelian groups $G$. For the non-abelian groups
$G$, the information that is missing is an eigenvalue $\lambda_{\gamma}$ for
all $\gamma\in G$. This problem is solved for the quaternionic groups $Q_{4n}$
by noting that it consists of the matrices $\pm S^{j}$ and $\pm US^{j}$ for
$j=1,\ldots,n$, and the latter all have eigenvalues
$\epsilon_{4},-\epsilon_{4}$. Eigenvalues for the elements of the binary
octahedral group can be gotten from straightforward computation.

###  4.3. Pairs of elliptic curves

The stratification by automorphism group for $A_{1}$, the moduli space of
elliptic curves, is given by the three groups $C_{2}$, $C_{4}$ and $C_{6}$.
The corresponding strata have Euler characteristics $-1$, $1$ and $1$
respectively. The two latter strata are points which can be represented by the
curves $y^{2}=x(x^{2}-1)$ and $y^{2}=x^{3}-1$ respectively. The automorphism
group is generated by the element $y\mapsto-y$ for $C_{2}$, by
$y\mapsto\epsilon_{4}y,x\mapsto-x$ for $C_{4}$ and
$y\mapsto-y,x\mapsto\epsilon_{3}x$ for $C_{6}$. For all elliptic curves of the
form $y^{2}=f(x)$, $H^{0}(C_{f},\Omega)$ has a basis consisting of the
differential $\omega_{0}=dx/y$. The eigenvalues of the induced action on
$H^{1}(C_{f},\mathbb{C})$ of the generators of the automorphism groups given
above then equals $-1,-1$ for $C_{2}$, $\epsilon_{4},-\epsilon_{4}$ for
$C_{4}$ and $\epsilon_{6},-\epsilon_{6}$ for $C_{6}$. The induced action of
the generators on the Weierstraß points (after choosing an ordering), which
correspond to the roots of $f(x)$ (together with infinity) equals
$\mathrm{id}$, $(12)$ and $(123)$ respectively.

Consider now $A_{1,1}\cong(A_{1}\times A_{1})/\mathfrak{S}_{2}$, which is the
moduli space of unordered pairs of elliptic curves. This has the consequence
that a pair of equal (or isomorphic) elliptic curves $E\times E$ will have an
extra automorphism that sends $(p_{1},p_{2})\in E\times E$ to
$(p_{2},p_{1})\in E\times E$. There will therefore be seven possible
automorphism groups for $A_{1,1}$, namely $C_{2}\times C_{2}$,
$C_{2}\wr\mathfrak{S}_{2}$, $C_{2}\times C_{4}$, $C_{2}\times C_{6}$,
$C_{4}\wr\mathfrak{S}_{2}$, $C_{4}\times C_{6}$ and
$C_{6}\wr\mathfrak{S}_{2}$, where $\wr$ denotes the wreath product. The Euler
characteristics of the corresponding strata are directly found to be
$1,-1,-1,-1,1,1$ and $1$, respectively. Take any two elliptic curves $E_{1}$
and $E_{2}$ with automorphism groups $G_{1}$ and $G_{2}$. Since
$H^{1}(E_{1}\times E_{2},\mathbb{C})\cong H^{1}(E_{1},\mathbb{C})\oplus
H^{1}(E_{2},\mathbb{C})$ it is straightforward, using the information for
$A_{1}$ above, to compute the action of $G_{1}\times G_{2}$ if $E_{1}$ and
$E_{2}$ are not isomorphic, and of $G_{1}\wr\mathfrak{S}_{2}$ if $E_{1}$ and
$E_{2}$ are isomorphic. The action of $G_{1}\times G_{2}$ (and of
$G_{1}\wr\mathfrak{S}_{2}$) on the six Weierstraß point that are distinct from
infinity on both elliptic curves is also straightforward.

##  5\. Isotypical decomposition in the vector-valued case

In this section we assume that $j>0$, so we are dealing with vector-valued
Siegel modular forms. As a consequence of [8, Proposition 1], we have

$M_{0,j}(\Gamma[2])=\left\\{0\right\\}\text{ for any }j>0.$

Theorem A.5 by G. Chenevier in [11] tells us that

$M_{1,j}(\Gamma[2])=S_{1,j}(\Gamma[2])=\left\\{0\right\\}\text{ for any }j>0.$

For $k=2$, there is no dimension formula for the space $M_{2,j}(\Gamma[2])$ in
general. A conjectural description of the isotypical decomposition of the
space $S_{2,j}(\Gamma[2])$ is given in [11, Conjecture 1.2]. As this
conjecture has only been verified for $j<12$ we decided to not implement the
isotypical decomposition of the space $M_{2,j}(\Gamma[2])$ in our code. For
$k=3$, the situation is also still conjectural but with more evidence. In fact
only the isotypical decomposition of $E_{c,\mathrm{Eis}}(A_{2}[2],V_{j,0})$
(see Theorem 5.3 to understand how this part contributes to the isotypical
decomposition of the space $S_{3,j}(\Gamma[2])$, and then further in Remark
5.4) is still conjectural so we decided to implement the isotypical
decomposition of the space $S_{3,j}(\Gamma[2])$ in our code. Evidence towards
this conjecture are given for example by the results of Petersen in [22] or in
Section 6.6 of [5]. We start by recalling the dimension formula for the space
$M_{k,j}(\Gamma[2])$ which can also be used to check its conjectural
isotypical decomposition for $k=3$.

######  Theorem 5.1 ([29, Theorems 2 and 3] and [12, Theorem 12.1]).

For $k\geqslant 3$ odd and $j\geqslant 2$ even we have

$\displaystyle\dim M_{k,j}(\Gamma[2])=\dim
S_{k,j}(\Gamma[2])=\frac{1}{24}\big{(}$ $\displaystyle
2(j+1)k^{3}+3(j^{2}-2j-8)k^{2}+(j^{3}-9j^{2}-42j+118)k$
$\displaystyle-2j^{3}-9j^{2}+152j-216\big{)}.$

For $k\geqslant 4$ even and $j\geqslant 2$ even we have

$\displaystyle\dim M_{k,j}(\Gamma[2])=\frac{1}{24}\big{(}$ $\displaystyle
2(j+1)k^{3}+3(j^{2}-2j+2)k^{2}+(j^{3}-9j^{2}-12j+28)k-2j^{3}-9j^{2}+182j-336\big{)}.$

######  Remark 5.2.

For $k=3$, this formula is rather pretty

$\dim S_{3,j}(\Gamma[2])=(j-2)(j-3)(j-4)/24.$

###  5.1. Isotypical decomposition of $M_{k,j}(\Gamma[2])$

For $k\geqslant 0$, the results of Rösner (see [23, Section 5]) and those of
[12, Section 13] tell us

$M_{k,j}(\Gamma[2])=E_{k,j}(\Gamma[2])\oplus
S_{k,j}(\Gamma[2])=E^{\textbf{(Q)}}_{k,j}(\Gamma[2])\oplus
S^{\textbf{(Y)}}_{k,j}(\Gamma[2])\oplus S^{\textbf{(G)}}_{k,j}(\Gamma[2])$

Theorem 5.13 and Remark 5.14 in [23] which prove Conjecture 6.4 of [4] give us
the isotypical decomposition of the space $S^{\textbf{(Y)}}_{k,j}(\Gamma[2])$
for $k\geqslant 3$ and $j>0$

$\displaystyle\dim_{\mathfrak{S}_{6}}S^{\textbf{(Y)}}_{k,j}(\Gamma[2])=\mu_{1}\,s[2^{3}]\,+\,\mu_{2}\,s[2,1^{4}]+\,\mu_{3}\,s[1^{6}]\quad\text{with}\quad\mu_{1}$
$\displaystyle=d^{+}_{2,j+2k-2}d^{+}_{2,j+2}+d^{-}_{2,j+2k-2}d^{-}_{2,j+2}$
$\displaystyle\mu_{2}$ $\displaystyle=d_{4,j+2k-2}d_{4,j+2}$
$\displaystyle\mu_{3}$
$\displaystyle=d^{+}_{2,j+2k-2}d^{-}_{2,j+2}+d^{-}_{2,j+2k-2}d^{+}_{2,j+2}$

where the integers $d_{N,k}$ and $d^{\pm}_{N,k}$ are defined as in (4).

Proposition 13.1 in [12] gives us $E^{\textbf{(Q)}}_{k,j}(\Gamma[2])={0}$ for
$k$ odd. For $k\geqslant 2$, this proposition (note that there is a typo in
[12]) tells us that

$\displaystyle\dim_{\mathfrak{S}_{6}}E^{\textbf{(Q)}}_{2k,j}(\Gamma[2])=$
$\displaystyle\text{Ind}_{H}^{\mathfrak{S}_{6}}\left(\text{{Sym}}^{j/2+k}\bigl{(}s[2,1]\bigr{)}-s[3]-s[2,1]\right)=\text{Ind}_{H}^{\mathfrak{S}_{6}}\left(S_{2k+j}(\Gamma(2))\right)$
$\displaystyle=$ $\displaystyle
d_{1,2k+j}(s[6]+s[5,1])+(2d_{1,2k+j}+d_{2,2k+j})s[4,2]+(d_{1,2k+j}+d_{2,2k+j})(s[3,2,1]+s[2^{3}])$
$\displaystyle+d_{4,2k+j}(s[3,1^{3}]+s[2,1^{4}])$

where the last identity followed from Section 3.2.1. So to get the isotypical
decomposition of the space $M_{k,j}(\Gamma[2])$ it remains to determine it for
the space $S^{\textbf{(G)}}_{k,j}(\Gamma[2])$. This is done in the next
section.

###  5.2. An isotypical dimension formula for
$S^{\textbf{(G)}}_{k,j}(\Gamma[2])$

First we introduce some notation from [4]. Let

$\begin{array}[]{ll}A=s[6]\oplus s[5,1]+s[4,2],&\qquad A^{\prime}=s[6]\oplus
s[4,2]\oplus s[2^{3}],\\\ B=s[4,2]\oplus s[3,2,1]+s[2^{3}],&\qquad
B^{\prime}=s[5,1]\oplus s[4,2]\oplus s[3,2,1],\\\ C=s[3,1^{3}]\oplus
s[2,1^{4}],&\qquad C^{\prime}=s[4,1^{2}]\oplus s[3^{2}].\end{array}$

For any $l,m$, with $l>m>0$, put $n=l+m+4$, $n^{\prime}=l-m+2$ and define

$\displaystyle E_{c,\mathrm{Eis}}(A_{2}[2],V_{l,m})=$
$\displaystyle(d_{1,n^{\prime}}-d_{1,n})\,(A^{\prime}+B^{\prime})+\,(d_{2,n^{\prime}}-d_{2,n})\,B^{\prime}+(d_{4,n^{\prime}}-d_{4,n})\,C^{\prime}+\frac{1}{2}\bigl{(}1+(-1)^{m}\bigr{)}\,(A+B)$
$\displaystyle+2\bigl{(}(d_{1,m+2}-d_{1,l+3})\,(A+B)+(d_{2,m+2}-d_{2,l+3})\,B+(d_{4,m+2}-d_{4,l+3})\,C\bigr{)}$
(10)

and

$\displaystyle E_{c,\mathrm{endo}}(A_{2}[2],V_{l,m})=$
$\displaystyle-2\Bigl{(}d_{4,n^{\prime}}\,\bigl{(}d_{4,n}\,s[3,1^{3}]+d_{1,n}\,s[3^{2}]+(d_{1,n}+d_{2,n})\,s[4,1^{2}]\bigr{)}$
$\displaystyle+d_{2,n^{\prime}}\,\bigl{(}(d_{1,n}+d_{2,n})\,s[3,2,1]+d_{4,n}\,s[4,1^{2}]+d_{1,n}\,s[4,2]+d_{1,n}\,s[5,1]\bigr{)}$
$\displaystyle+d^{+}_{2,n^{\prime}}\,\bigl{(}d^{+}_{2,n}\,s[4,2]+d^{-}_{2,n}\,s[5,1]\bigr{)}+d^{-}_{2,n^{\prime}}\,\bigl{(}d^{-}_{2,n}\,s[4,2]+d^{+}_{2,n}\,s[5,1]\bigr{)}$
$\displaystyle+d_{1,n^{\prime}}\,\bigl{(}d_{1,n}\,(A^{\prime}+B^{\prime})+d_{2,n}\,B^{\prime}+d_{4,n}\,C^{\prime}\bigr{)}\Bigr{)}$

as elements of the representation ring $\mathbb{Z}[\mathfrak{S}_{6}]$.

######  Theorem 5.3.

For any $k\geq 4$ and $j>0$, put $l=j+k-3$ and $m=k-3$. Then

$\mathrm{dim}_{\mathfrak{S}_{6}}S^{\textbf{(G)}}_{k,j}(\Gamma[2])=-\frac{1}{4}\Bigl{(}E_{c}(A_{2}[2],V_{l,m})-E_{c,\mathrm{Eis}}(A_{2}[2],V_{l,m})-E_{c,\mathrm{endo}}(A_{2}[2],V_{l,m})+2\,\mathrm{dim}_{\mathfrak{S}_{6}}S^{\textbf{(Y)}}_{k,j}(\Gamma[2])\Bigr{)}.$

###### Proof.

In [4], the compactly supported $\ell$-adic Euler characteristic of local
systems $\mathbb{V}_{l,m}$ on $\mathcal{A}_{2}[2]$ taking values in the
Grothendieck group of (absolute) Galois representations is decomposed into the
following pieces,

$e_{c}(\mathcal{A}_{2}[2],\mathbb{V}_{l,m})=e_{c,\mathrm{Eis}}(\mathcal{A}_{2}[2],\mathbb{V}_{l,m})+e_{c,\mathrm{endo}}(\mathcal{A}_{2}[2],\mathbb{V}_{l,m})-S[l-m,m+3,\Gamma[2]].$

The formula for $E_{c,\mathrm{Eis}}(A_{2}[2],V_{l,m})$ (respectively
$E_{c,\mathrm{endo}}(A_{2}[2],V_{l,m})$) is found by taking dimensions in the
formula for $e_{c,\mathrm{Eis}}(\mathcal{A}_{2}[2],\mathbb{V}_{l,m})$
(respectively $e_{c,\mathrm{endo}}(\mathcal{A}_{2}[2],\mathbb{V}_{l,m})$) in
[4, Theorem 4.4] (respectively [4, Conjecture 7.1]). The representation
$S[l-m,m+3,\Gamma[2]]$ should conjecturally consist of $2$-dimensional pieces
for each Hecke eigenvector in $S^{\textbf{(Y)}}_{k,j}(\Gamma[2])$, with
isotypic decomposition given in [4, Conjecture 6.4], and $4$-dimensional
pieces for each Hecke eigenvector in $S^{\textbf{(G)}}_{k,j}(\Gamma[2])$.

The conjectural description in [4] described above, has been proven in [23].
Conjectures 7.1 and 6.4 of [4] are proved by Theorem 5.13 of [23], see Remark
5.14 of [23]. The result then follows from [23, Corollary 5.20]. ∎

######  Remark 5.4.

In [4], directly after Theorem 4.4, it is conjectured that
$E_{c,\mathrm{Eis}}(A_{2}[2],V_{l,0})$ for any $l>0$ is given by (5.2), with
the difference that one needs to put $d_{1,2}=-1$. If we assume this
conjecture to be true, and we define $E_{c,\mathrm{endo}}(A_{2}[2],V_{l,0})$
for any $l>0$ using the formula above, then Theorem 5.3 also holds for $k=3$
and $j>0$ using the same proof (and the same results of [23]).

## Acknowledgement

The second author was supported by the Simons Foundation Award 546235 at the
Institute for Computational and Experimental Research in Mathematics at Brown
University. We thank Eran Assaf and Gerard van der Geer for useful
discussions.

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