Siegel Modular Forms of Degree 2 and 3
Modular forms on the full level 2 congruence subgroup $\Gamma_2[2]$ of $\Sp(4,Z)= \ker\bigl(\Sp(4,\Z) \to \Sp(4,\Z/2\Z)\bigr)$

We denote the space of cusp forms on $\Gamma_2[2]$ of weight $(j,k)$, i.e. corresponding to $\Sym^j$ tensor $\det^k$, by $S_{j,k}[2]$. The symmetric group $S_6=\Sp(4,\F_2)$ acts on the spaces $S_{j,k}[2]$.

The irreducible representations of the symmetric group $S_6=\Sp(4,\F_2)$ are given as $s[P]$ with $P$ running through partitions of 6: $s[6],\, s[5,1],\, s[4,2],\, s[4,1^2],\, s[3^2],\, s[3,2,1],\, s[3,1^3],\, s[2^3],\, s[2^2,1^2],\, s[2,1^4],\, s[1^6]$.

For the actual isomorphism see [2]. To fix things, the space of scalar-valued modular forms of weight 2 is the irrep $s[2^3]$.

For given $(j,k)$ with $k > 2$ and $j+2k \leq 100$ you will find:

  1. the isotypical decomposition of the space $S_{j,k}[2]$ of cusp forms of weight $(j,k)$ on the level 2 congruence subgroup $\Gamma_2[2]$ of $\Sp(4,\Z)$
  2. the isotypical decomposition of the subspace of lifts of $S_{j,k}[2]$
  3. the isotypical decomposition of the subspace of non-lifts of $S_{j,k}[2]$
  4. the motivic form of the lifts for weight $(j,k)$
  5. for an odd prime power $q < 200$ a vector of traces of the Hecke operator $T(q)$ on the isotypical subspaces of $S_{j,k}[2]$
  6. for an odd prime $p$ and a character $s[P]$ of $S_6$ the spinor $L$-factor on the $s[P]$ subspace of $S_{j,k}[2]$ of cusp forms of weight $(j,k)$ on $\Gamma_2[2]$ if available

For 5) note that our convention for $T(q)$ for $q=p^a$ with $a>1$ is different from the usual one; the Hecke operator $T(p)$ with $p$ prime is the same as the usual one. For $q=p^a$ with $a>1$ our $T(q)$ is different from the usual $T(q)$; the eigenvalues of our $T(q)$ are described in terms of Satake parameters in Definition 10.1 in the paper [3]. For notation of 4) we refer to [3], see also here. See Section 7 in [2] for a description of the lifted forms. Here one finds information on reducible characteristic polynomials.

If you wish to have all the data in a single file, please contact one of the initiators.

The data are based on our computer counts and the conjectural formulas of [2] that are proven in [6].

Look up data

There is no data available for these values of $j$ and $k$.