On the average size of the eigenvalues of the Hecke operators

TL;DR

This paper investigates the average magnitude of eigenvalues of Hecke operators on cuspidal modular forms, using quadratic mean to measure their size from both vertical and horizontal viewpoints.

Contribution

It provides a detailed analysis of the average size of Hecke eigenvalues in modular forms spaces from two different perspectives.

Findings

Quantifies the average size of eigenvalues using quadratic mean.

Analyzes eigenvalues in both vertical and horizontal contexts.

Offers insights into the distribution of Hecke eigenvalues.

Abstract

We determine the average size of the eigenvalues of the Hecke operators acting on the cuspidal modular forms space $S_k(\Gamma_0(N))$ in both the vertical and the horizontal perspective. The "average size" is measured via the quadratic mean.