TITLE: Field generated by the Fourier coefficients of a modular form
QUESTION [6 upvotes]: Let $f = \sum_n a_n q^n$ be a cuspidal newform of weight $k$ on $\Gamma_0(N)$ for some $N$.  Let $K_f$ be the number field generated by the $a_q$ as $q$ runs over all primes.
My question: if we consider the field generated by all but one of these $a_q$, can this field be smaller than $K_f$?  
(edited based on Sawin's answer to only look at Fourier coefficients at primes)

REPLY [12 votes]: No, strong multiplicity one says that all-but-finitely-many of the $a_p$'s determine all the others.
Edit in response to comment/query, and further in response to subsequent comments: ... and, once all the other coefficients are determined by strong multiplicity one (for newforms), invoke Shimura's results (arguably going back to Fricke-Klein in principle) that automorphisms of the complex numbers over $\mathbb Q$ (not merely Galois automorphisms on an algebraic closure of $\mathbb Q$) can be applied to the Fourier coefficients of a holomorphic modular form, producing another. In the case at hand, applying such an automorphism over the field generated by all the given Fourier coefficients, and subtracting, would either give a modular form with only finitely-many non-zero $a_p$'s (impossible), or $0$ (giving the desired result).
[Thanks to MF1 for corrections.]