The Fourier expansion of Hecke operators for vector-valued modular forms

Abstract

We compute the Fourier expansion of Hecke operators on vector-valued modular forms for the Weil representation associated to a lattice $L$. The Hecke operators considered in this paper include operators $T(p^{2l})$ where $p$ is a prime dividing the level of the lattice $L$. Additionally, an explicit formula for a general type of Gauss sum associated to a lattice $L$ drops out as a by-product.