  Let $\phi$ denote a primitive Hecke-Maass cusp form for $\Gamma_o(N)$ with
the Laplacian eigenvalue $\lambda_\phi=1/4+t_{\phi}^2$. In this work we show
that there exists a prime $p$ such that $p\nmid N$, $|\alpha_{p}|=|\beta_{p}| =
1$, and $p\ll(N(1+|t_{\phi}|))^c$, where $\alpha _{p},\;\beta _{p}$ are the
Satake parameters of $\phi$ 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 $34/35$.

Number Theory