An omega result for the first sign change of coefficients of symmetric power $L$-functions of Hecke-Maass cusp forms

Abstract

Recently, Lamzouri proved a lower bound for the least positive integer $n_f$ for which the Hecke eigenvalue $\lambda_f(n_f)<0$, showing it satisfies $n_f \ge (\log k)^{1-o(1)}$ for many holomorphic Hecke cusp forms of even integral weight $k$. In this paper, we extend Lamzouri's result to Hecke-Maass forms without assuming the Generalized Ramanujan Conjecture, and further prove that similar results also hold for the coefficients of symmetric power $L$-functions of Hecke-Maass forms.