The cusp forms of weight $3$ on $\Gamma _ 2(2,4,8)$

Abstract:

The congruence subgroup ${\Gamma _2}(2,4,8)$ of the group ${\Gamma _2}$ of $4 \times 4$ integral symplectic matrices is contained in ${\Gamma _2}(4)$ and contains ${\Gamma _2}(8)$, with ${\Gamma _2}(n)$ the principal congruence subgroup of level n. The Satake compactification of the quotient of the three-dimensional Siegel upper half space by ${\Gamma _2}(2,4,8)$ is shown to be a complete intersection of ten quadrics in ${\mathbb {P}^{13}}$. We determine the space of global holomorphic three forms on this space, which coincides with the space of cusp forms of weight 3 on ${\Gamma _2}(2,4,8)$; it has dimension 2283. Finally, we study the action of the Hecke operators on this space and consider the Andrianov L-functions of some eigenforms.