Asymptotics and sign patterns of Hecke polynomial coefficients

TL;DR

This paper analyzes the asymptotic behavior and sign patterns of Hecke polynomial coefficients, providing new insights into their properties for large levels or weights and confirming a conjecture on their non-vanishing.

Contribution

It establishes the asymptotic behavior of Hecke polynomial coefficients and verifies a conjecture on their non-vanishing for large parameters.

Findings

Determined the asymptotic behavior of Hecke polynomial coefficients.

Identified sign patterns for large levels or weights.

Verified a conjecture on the non-vanishing of coefficients in most cases.

Abstract

We determine the asymptotic behavior of the coefficients of Hecke polynomials. In particular, this allows us to determine signs of these coefficients when the level or the weight is sufficiently large. In all but finitely many cases, this also verifies a conjecture on the nanvanishing of the coefficients of Hecke polynomials.