Siegel Eisenstein series (awaiting review)

The (Siegel) Eisenstein series in degree $n$ of even weight \(k>n+1\) is defined by the formula \[ \psi_k(Z)=\sum_{(C,D)}\det(CZ+D)^{-k}, \] where the summation extends over all inequivalent bottom rows \((C,D)\) of elements of the integral symplectic group \({\rm Sp}(2n,\mathbb{Z})\) with respect to left multiplication by elements of \({\rm GL}(n,\mathbb{Z})\).

Siegel Eisenstein series are also defined for some subgroups of the full integral symplectic group as well.