Hecke Operators for Maass Waveforms on PSL(2,Z) with Integer Weight and Eta Multiplier

TL;DR

This paper develops Hecke operators for Maass waveforms with non-zero integer weight and eta multiplier, deriving new multiplicativity relations for Fourier coefficients that extend classical results.

Contribution

It introduces Hecke operators for Maass waveforms with non-trivial multipliers and establishes generalized multiplicativity relations for their Fourier coefficients.

Findings

Derived multiplicativity relations for Fourier coefficients

Established a new relation between positive and negative index coefficients

Provided numerical examples illustrating the relations

Abstract

We construct Hecke operators acting on Maass waveforms of integer non-zero weight and transforming according to a non-trivial multiplier system on the modular group. Using these Hecke operators we obtain multiplicativity relations for the Fourier coefficients of such Maass waveforms. These relations generalize the usual weight zero relations. We also obtain an unexpected relation between coefficients with positive and negative indices with a constant of proportionality involving the Laplace eigenvalue. Numerical examples of multiplicativity relations are given at the end of the paper.