Let $N$ and $p$ be relatively prime integers with $p$ a prime. Suppose $f$ is a weight $k=2$ (normalized, cuspidal, etc) newform of level $\Gamma_1(N) \cap \Gamma_0(p)$. I seem to recall the existence of a theorem to the following effect:
Let $\rho_{f,p}: G_{\mathbf{Q}_p} \to \mathrm{GL}_2(\bar{\mathbf{F}}_p)$ be the mod $p$ $p$-adic representation associated to $f$ (and some prime ${\frak{p}} \mid p$). Suppose $\rho_{f,p}$ is irreducible and finite flat. Then there exists a newform $g$ of weight $k=2$ and level $\Gamma_1(N)$ which is congruent to $f$ mod $p$.
Does such a theorem actually exist? The content I'm looking for is the ability to simultaneously remove $p$ from the level while also remaining in weight $k=2$.
Theorem 2.1 of these notes attributes a similar theorem to Mazur, but it is required that $f$ be of level $\Gamma_0(Np)$. Can this be strengthened to allow nontrivial nebentype away from $p$?
