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:

https://mathoverflow.net/questions/131604/field-generated-by-the-fourier-coefficients-of-a-modular-form

and

https://mathoverflow.net/questions/114145/a-sentence-in-shimuras-on-the-periods-of-modular-forms/114170

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