On the Hecke eigenvalues of Maass forms

Let $\phi$ denote a primitive Hecke-Maass cusp form for $\Gamma_0(N)$ with the Laplacian eigenvalue $\lambda_\phi={1\over 4}+t_{\phi}^2$. In~this~work we show that there exists a prime $p$ such that {$p\!\not|\ N$,} $|\alpha_{p}|=|\beta_{p}|=1$, and $p\ll(N(1+|t_{\phi}|))^c$, where $\{\alpha_p,\,\beta_p\}$ is the Satake parameter of $\phi$ at $p$, and $c$ is an absolute constant with $0<c<1$. In fact, $c$ can be taken as $8/11+\epsilon$ (or even 0.27331 by a more elaborate numerical calculation). In addition, we prove that the natural density of such primes $p$ ($p\!\not|\ N$ and $|\alpha_{p}|=|\beta_{p}| = 1$) is at least $34/35$.