Let $K\_f$ denote the number field generated by the Fourier coefficients $a\_n$ of a normalized primitive holomorphic cusp form $f$.
On page 2, line 6 of the paper mentioned in the title, Shimura writes that $K\_f$ is generated by $a\_p$ for almost all primes $p$. In the next sentence, he says that it follows trivially from the Multiplicity one theorem. 


I don't see how it follows. I shall appreciate any comment.



A sentence in Shimura's "On The Periods of Modular Forms"
Suppose that a co-finite subset of the $a\_p$'s generate a field $L$. Applying any element $\sigma$ of the absolute Galois group of $L$ to $f$ yields another form $g$ whose Fourier coefficients $b\_p$ are equal to $a\_p$ for all but finitely many $p$. By strong multiplicity one, $g=f$ and $b\_p = a\_p$ for all $p$. Therefore $\sigma$ fixes $K\_f$ as well, and since $L \subset K\_f$ it follows that $L=K\_f$.


