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Questions about modular forms and related areas

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Does the $p$-part of the level of a newform appear in its attached $p$-adic representation?

Let $f$ a newform of weight $2$ on $\Gamma_0(Np^r)$, $N$ coprime to $p$, and consider its $p$-adic Galois representation $$ \rho:G_{\mathbb Q}\longrightarrow GL_2(\bar{\mathbb Q}_p) $$ It's a theorem …
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