Field generated by the Fourier coefficients of a modular form 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)
 A: No. Using all but one of the coefficients, it is easy to compute the Hecke operator eigenvalues using just addition, subtraction, multiplication, and division. Then one can compute the missing coefficient using a Hecke operator.
A: 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.] 
