Quadratic base change and resonance sums for holomorphic cusp forms on \(\Gamma_0(N)\) (Q6585296)

Let \(S_X(\alpha, \beta, \pi)\) be a resonance sum 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 \(\sqrt{D}\) for some \(D\) and \(N\) with \(k\) is even. Then, in the paper under review, the author expresses \(S_X(\alpha, \beta, \pi)\) in terms of the Meier-G function. To do this, he uses the Voronoi summation formula. As a second contribution, he determines the asymptotic behavior of that resonance sum when \(X\) tends to infinity by the same argument. Finally, he proves that by 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.