Let $K$ be an imaginary quadratic field. Let $f \in S_2(\mathfrak{n})$ be a weight $2$ cuspidal cof level $\Gamma_0(\mathfrak{n})$ over $K$ (for definitions one can see http://www.lmfdb.org/knowledge/show/mf.bianchi.bianchimodularforms ), consider its $L$-function

$L(f,s)=\sum_{\mathcal{P} \not =0 \in Spec(O_K)} c(\mathcal{P}) (N\mathcal{P})^{-s}$

here $c(\mathcal{P})$ are the Fourier coefficients at the cusp $\infty$ and we assume they are all rational numbers.

The question is, do we know that $c(\mathcal{P})=0$ for infinitely many prime ideals $\mathcal{P}$? How about $c(\mathcal{P})=0 \mod N(\mathcal{P})$?

Motivation: if we are considering similar problems over $\mathbb Q$, then we know $a_p=0$ for infinitely many $p$ because the corresponding elliptic curve over $\mathbb Q$ must have supersingular reduction at infinitely many primes.

Edit: as the answer shows, we shall expect $c(\mathcal{P})=0 \mod p$ happens for infinitely many times, rather than $\mod N(\mathcal{P})$.