The quantitative distribution of Hecke eigenvalues of Maass forms

TL;DR

This paper provides quantitative analysis of the distribution of Hecke eigenvalues for Maass forms, with applications to sign change results for these eigenvalues and their symmetric squares.

Contribution

It introduces new quantitative results on Hecke eigenvalue distribution and applies them to sign change phenomena for Maass forms and their symmetric squares.

Findings

Quantitative distribution results for Hecke eigenvalues

Applications to sign change and Omega results

Insights into symmetric square eigenvalues

Abstract

Let $f$ be a normalized Hecke-Maass cusp form of weight zero for the group $SL_2(\mathbb Z)$. This article presents several quantitative results about the distribution of Hecke eigenvalues of $f$. Applications to the $\Omega_{\pm}$-results for the Hecke eigenvalues of $f$ and its symmetric square sym$^2(f)$ are also given.