Explicit Action of Hecke Operators on Siegel Modular Forms Original Research Article

We develop an algorithm for determining an explicit set of coset representatives (indexed by lattices) for the action of the Hecke operators T(p), Tj(p2) on Siegel modular forms of fixed degree and weight. This algorithm associates each coset representative with a particular lattice Ω, pΛsubset of or equal toΩsubset of or equal toimage Λ where Λ is a fixed reference lattice. We then evaluate the action of the Hecke operators on Fourier series. Since this evaluation yields incomplete character sums for Tj(p2), we complete these sums by replacing this operator with a linear combination of Tℓ(p2), 0less-than-or-equals, slantℓless-than-or-equals, slantj. In all cases, this yields a clean and simple description of the action on Fourier coefficients.