Let $f=\sum\_{n\ge1} a\_nq^n$ be an Hecke eigenform cusp form of integral weight $k$ on $\Gamma\_0(N)$ with character $\varepsilon\pmod N,$ and let $M\_f$ be the subfield of $\mathbb{C}$ generated by the image of $\varepsilon$ and all $a\_p$ ($p$ runs on primes) 
$$M\_f:=\mathbb{Q}(\{a\_p\}\_p,\varepsilon)$$
On what condition we have $\dfrac{a\_p}{p^{(k-2)/2}}\in M\_f$ ?



The subfield of $\mathbb{C}$ generated by Fourier coefficients of prime index
When $k$ is even the answer is *always* since $p^{(k-2)/2} \in \mathbb Q$.
When $k$ is odd the answer is *never* since if this was true, then for all $p$ (prime not divdidig $N$), we would have $p^{1/2} \in M\_f$, so $M\_f$ would contain the field of infinite degree over $\mathbb Q$ generated by the square roots of those prime, contradicting the fact that $M\_f$ is a finite extension of $\mathbb Q$.


