An omega result for the least negative Hecke eigenvalue

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 $n_f\ge (\log k{)}^{1-o(1)}\mathrm{.}$ 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 $(\log k{)}^{1\mathrm{/}2+o(1)}$. We also discuss an extension of our result to primitive holomorphic cusp forms of weight $k$ and squarefree level $N\ge 1$.