Vorono\"{i} summation via switching cusps

We consider the Fourier expansion of a Hecke (resp.\ Hecke–Maa\ss) cusp form of general level $N$ at the various cusps of ${\mathrm{\Gamma }}_0(N)\text{bs}\text{Hb}$. 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.