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 $N\geq 1$.
From: Youness Lamzouri [view email]
[v1]
Mon, 9 Feb 2026 18:33:37 UTC (8 KB)
[v2]
Mon, 1 Jun 2026 20:39:37 UTC (9 KB)
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