Voronoï summation via switching cusps

Abstract: We consider the Fourier expansion of a Hecke (resp.\ Hecke--Maa\ss) cusp form of general level $N$ at the various cusps of $\Gamma_{0}(N)\bs\Hb$. We explain how to compute these coefficients via the local theory of $p$-adic Whittaker functions and establish a classical Vorono\"i summation formula allowing an arbitrary additive twist. Our discussion has applications to bounding sums of Fourier coefficients and understanding the (generalised) Atkin--Lehner relations.