Universal Fourier expansions of Bianchi modular forms

TL;DR

This paper extends the concept of universal Fourier expansions from classical modular forms to Bianchi modular forms over Euclidean imaginary quadratic fields, utilizing Hecke operators and modular symbols.

Contribution

It introduces a generalized framework for Fourier expansions of Bianchi modular forms, building on prior work and specific computational techniques.

Findings

Established universal Fourier expansions for Bianchi modular forms.

Computed Hecke operator actions on Manin symbols.

Extended Merel's work to new algebraic settings.

Abstract

We generalize Merel's work on universal Fourier expansions to Bianchi modular forms over Euclidean imaginary quadratic fields, under the assumption of the nondegeneracy of a pairing between Bianchi modular forms and Bianchi modular symbols. Among the key inputs is a computation of the action of Hecke operators on Manin symbols, and building upon the Heilbronn-Merel matrices constructed by Mohamed.