Modular forms on $\Sp(4,\Z)$

We denote the space of cusp forms on $\Sp(4,\Z)$ of weight $(j,k)$, i.e. corresponding to $\Sym^j$ tensor $\det^k$, by $S_{j,k}$.

For chosen weight $(j,k)$ with $k \geq 3$ and $j+2k \leq 100$ you can click "look up" to get the following data

1. the dimension of the space $S_{j,k}$ of cusp forms of weight $(j,k)$ on $\Sp(4,\Z)$
2. the dimension of the space of lifted forms of weight $(j,k)$
3. the dimension of the space of non-lifted forms of weight $(j,k)$
4. the lifts (motivic form) of weight $(j,k)$
5. for a prime power $q\leq 199$ the trace of the Hecke operator $T(q)$ on the space of non-lifted forms in $S_{j,k}$
6. for a prime $p$ with $2 \leq p \leq 13$ the spinor $L$-factor on the space of non-lifted forms in $S_{j,k}$ if this space is 1-dimensional.

For 5) note that our convention for $T(q)$ for $q=p^a$ with $a>1$ is different from the usual one; we use the notation from the paper [3], Section 10, where the relation with the usual definition of $T(q)$ is explained. In 4) we mean Saito-Kurokawa lift, for notation we refer to [3]. In particular $S[k]$ stands for the motive of cusp forms of weight $k$ on $\SL(2,\Z)$.

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

The data here rest on the conjectures from [1], see also [3], and these formulas have been proven to hold by work of Weissauer [4] and Petersen [5]. Here one finds information on reducible characteristic polynomials.

### Look up data

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