Let $f = \sum_n a_n q^n$ be a cuspidal newform of some weight and level. Here I want to view the $a_n$ of $p$-adic numbers (by embedding $\overline{{\bf Q}}_p$ in ${\bf C}$ in some way).

Let $k_f$ denote the finite field generated by the reduction mod $p$ of all of the $a_q$ as $q$ varies over all primes. Let $k'_f$ denote the same field except that we omit a single $a_q$ for one prime $q$.

My question: can $k'_f$ be strictly smaller than $k_f$?

(Compare to the characteristic 0 analogue in:

Field generated by the Fourier coefficients of a modular form

and

A sentence in Shimura's "On The Periods of Modular Forms"

where strong multiplicity one guarantees that the field is unchanged if finitely many $a_q$ are deleted.)