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We find nice representatives for the 0-dimensional cusps of the degree $n$Siegel upper half-space under the action of $\Gamma_0(\stufe)$. To each ofthese we attach a Siegel Eisenstein series, and then we make explicit a resultof Siegel, realizing any integral weight average Siegel theta series ofarbitrary level $\stufe$ and Dirichlet character $\chi_L$ modulo $\stufe$ as alinear combination of Siegel Eisenstein series.

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