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

We find nice representatives for the 0-dimensional cusps of the degree n Siegel upper half-space under the action of (Gamma _0(mathcal N )). 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 (mathcal N ) and Dirichlet character (chi _{_L}) modulo (mathcal N ) as a linear combination of Siegel Eisenstein series.