On the occurrence of Hecke eigenvalues in sectors

TL;DR

This paper demonstrates that certain automorphic representations have Hecke eigenvalues with positive density in sectors of the complex plane, specifically for sectors with angles at least 2.63 radians.

Contribution

It establishes a lower bound on the angular sector size for which automorphic representations exhibit positive density of Hecke eigenvalues.

Findings

Hecke eigenvalues are densely distributed in sectors of the complex plane.

Positive density of eigenvalues is guaranteed for sectors with angles ≥ 2.63 radians.

The result applies to non-self-dual, non-solvable polyhedral type automorphic representations.

Abstract

Let $\pi$ be a non-self-dual unitary cuspidal automorphic representation of non-solvable polyhedral type for GL(2) over a number field. We show that $\pi$ has a positive upper Dirichlet density of Hecke eigenvalues in any sector whose angle is at least 2.63 radians.