Explicit calculations with Eisenstein series

Abstract

We find explicit change-of-basis formulas between Eisenstein series attached to cusps, and newform Eisenstein series attached to pairs of primitive Dirichlet characters. As a consequence, we prove a Bruggeman–Kuznetsov formula for newforms of square-free level and trivial nebentypus.We also derive, in explicit form, the Fourier expansion, Hecke eigenvalues, Atkin–Lehner pseudo-eigenvalues, and other data associated to these Eisenstein series, with arbitrary integer weight, level, and nebentypus.