Quadratic base change and resonance sums for holomorphic cusp forms on Γ0(N)

Let $D,k$ be integers with D square free and k even. Let N be a positive integer so that $(N,D)=1$ when D has residue one modulo four and $(N,4D)=1$ when D has residue two or three modulo four. In this paper the asymptotic behavior of a resonance sum $S_X(\alpha ,\beta ;\pi )$ attached to the quadratic base change lift of a holomorphic cusp form f of level N and weight k over the quadratic extension generated by $\sqrt{D}$ is computed. First a Voronoi summation formula is derived that expresses $S_X(\alpha ,\beta ;\pi )$ in terms of the Meier-G function. Then, using the known asymptotics of the Meier-G function the asymptotic behavior of $S_X(\alpha ,\beta ;\pi )$ as X approaches infinity is determined. It is then shown that using only finitely many Fourier coefficients of the form, one can recover the weight k and the level N, which is a special case of the multiplicity one theorem.