TITLE: Question about Fourier coefficients of a newform at primes
QUESTION [5 upvotes]: For $q:=e^{2\pi i z},$  let $f(z)=\sum_{n\ge 1}\lambda(n)n^{(k-1)/2}q^n$ be a normalized newform of type $(k,\chi)$  and level $N$. For any prime $p,$ we have
$$\lambda(p)=2\cos(\theta_p)\;\;\;\text{for some}\;\;\; \theta_p\in[0,\pi]$$ 
I wonder: Is $\theta_p\pmod{2\pi}$ take only finitely many values when $p$ run over primes number?

REPLY [7 votes]: No, as this would contradict the Sato-Tate conjecture. For CM forms this conjecture is easier and has been long known, see e.g. this MO entry.