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)



Field generated by the Fourier coefficients of a modular form
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.] 


