On the Hecke Eigenvalues of Maass Forms

Abstract

Let $\varphi$ denote a primitive Hecke-Maass cusp form for ${\Gamma }_o(N)$ with the Laplacian eigenvalue ${\lambda }_{\varphi }=1/4+t_{\varphi }^2$. In this work we show that there exists a prime $p$ such that $p\nmid N$, $|{\alpha }_p|=|{\beta }_p|=1$, and $p\ll (N(1+|t_{\varphi }|){)}^c$, where ${\alpha }_p,{\beta }_p$ are the Satake parameters of $\varphi$ at $p$, and $c$ is an absolute constant with $0<c<1$. In fact, $c$ can be taken as $0.27332$. In addition, we prove that the natural density of such primes $p$ ($p\nmid N$ and $|{\alpha }_p|=|{\beta }_p|=1$) is at least $34/35$.

Version 2: typos corrected and a new section on natural density added