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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.

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up vote 18 down vote accepted

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 $L=K_f$.

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