Level 1

The data provided here are conditional based on the conjectures made in [3].

We denote the space of cusp forms on ${\rm Sp}(6,\Z)$ of weight $(j,k,l)$ by $S_{j,k,l}$. Here the weight $(j,k,l)$ refers to the irreducible representation of ${\rm GL}(3)$ of highest weight $(j+k+l,k+l,l)$; scalar-valued modular forms are of weight $(0,0,l)$.

For chosen weight $(j,k,l)$ with $l\geq 4$ and $j+2k+3l-6 \leq 80$ you can click "look up" to get the following data:

1. the dimension of the space $S_{j,k,l}$ of cusp forms on ${\rm Sp}(6,{\Z})$
2. the dimension of the space of lifted forms of weight $(j,k,l)$
3. the dimension of the space of non-lifted forms
4. the motivic form of the lifted forms
5. the traces of the Hecke operators for the prime powers in the list $[2,3,4,5,7,8,9,11,13,16,17,19,23,25]$
6. in case the space of non-lifted cusp forms has dimension $1$, the spinor $L$-factor on the space of non-lifted forms for $p=2$
7. in case the space of non-lifted cusp forms has dimension $1$, a list of forms that are possibly lifts from $G_2$, see [7].

For 5) note that our convention for $T(q)$ 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. For the meaning of the space of lifts we also refer to the paper [3].

Look up data