Explicitly realizing average Siegel theta series as linear combinations of Eisenstein series

Abstract

We find nice representatives for the 0-dimensional cusps of the degree $n$ Siegel upper half-space under the action of $Γ_0(\stufe)$. To each of these we attach a Siegel Eisenstein series, and then we make explicit a result of Siegel, realizing any integral weight average Siegel theta series of arbitrary level $\stufe$ and Dirichlet character ${\chi }_L$ modulo $\stufe$ as a linear combination of Siegel Eisenstein series.