Murmurations of Maass forms

TL;DR

This paper demonstrates the existence of murmurations in Maass forms, revealing correlations between parity and Hecke eigenvalues as the eigenvalue parameter increases, advancing understanding of automorphic forms.

Contribution

It establishes the presence of murmurations in Maass forms, a novel phenomenon linking eigenvalues and Hecke eigenvalues in this family.

Findings

Existence of murmurations in Maass forms proven.

Correlations between parity and Hecke eigenvalues identified.

Behavior observed as eigenvalue parameter tends to infinity.

Abstract

We prove the existence of murmurations in the family of Maass forms of weight 0 and level 1 with their Laplace eigenvalue parameter going to infinity (i.e., correlations between the parity and Hecke eigenvalues at primes growing in proportion to the analytic conductor).