An omega result for the least negative Hecke eigenvalue

TL;DR

The paper proves that for many holomorphic Hecke eigenforms of large weight, the smallest integer with a negative eigenvalue grows nearly as fast as a power of the logarithm of the weight, improving previous bounds.

Contribution

It establishes a near-optimal lower bound on the least integer with a negative Hecke eigenvalue for large weight forms, extending previous prime-based results.

Findings

Existence of many forms with large least negative eigenvalue index

Lower bound on n_f is approximately (log k)^{1-o(1)}

Improves previous prime-based bounds for eigenvalue negativity

Abstract

We establish the existence of many holomorphic Hecke eigenforms $f$ of large weight $k$ for the full modular group, for which the least positive integer $n_f$ such that $\lambda_f(n_f)<0$ satisfies $n_f \ge (\log k)^{1-o(1)}.$ This is believed to be best possible up to the $o(1)$ term in the exponent, and improves on a result of Kowalski, Lau, Soundararajan and Wu, who showed that, when restricted to primes, the least prime $p$ such that $\lambda_f(p)<0$ can be as large as $(\log k)^{1/2+o(1)}$. We also discuss an extension of our result to primitive holomorphic cusp forms of weight $k$ and squarefree level $N\geq 1$.