Let $f = \sum_{n=1}^{\infty} a_n(f) q^n$ be a normalized eigenform in $S_2(\Gamma_0(N))^{\mathrm{new}}$, i.e., $f$ is a newform of level $N \geq 2$. Assume that $f$ does not have complex multiplication. Let $K_f$ be the number field generated by these Fourier coefficients. For every prime $p$, fix a prime $\mathcal{P}$ above $p$ in $\mathcal{O}_{K_f}$, the integral closure of $\mathbb{Z}$ in $K_f$ .

Is it possible that $$a_p(f) \equiv 1 \mod \mathcal{P}$$ for all primes $p \gg 0$?

For example, if one knows the existence of infinitely many non-ordinary primes for $f$, then the above congruence will not hold. This is the case when $K_f = \mathbb{Q}$, by the result of Elkies.