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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 $p$?

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  • $\begingroup$ i think if rhobar is tre ramifie then it is not going to work, because rhobar will not have a crystalline lift. $\endgroup$ – peliukas May 23 '15 at 20:45
  • $\begingroup$ The assumption that rhobar is finite flat is equivalent to it being peu (and not tres) ramifie $\endgroup$ – Jeff H May 24 '15 at 18:09
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The answer is yes. This result (and more) is contained in Theorem 6.4 of Diamond's "The refined conjecture of Serre".

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