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?



Question about Fourier coefficients of a newform at primes
No, as this would contradict the [Sato-Tate conjecture](https://en.wikipedia.org/wiki/Sato%E2%80%93Tate_conjecture). For CM forms this conjecture is easier and has been long known, see e.g. [this MO entry](https://mathoverflow.net/questions/114062/sato-tate-conjecture-for-cm-modular-forms).


