Explicit calculations with Eisenstein series

TL;DR

This paper derives explicit formulas connecting different Eisenstein series and provides detailed data such as Fourier expansions and eigenvalues for these series across various weights, levels, and characters.

Contribution

It introduces explicit change-of-basis formulas between Eisenstein series and newform Eisenstein series, and proves a Bruggeman-Kuznetsov formula for square-free level newforms.

Findings

Explicit change-of-basis formulas between Eisenstein series

A proven Bruggeman-Kuznetsov formula for square-free level newforms

Explicit Fourier expansions and eigenvalues for 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.