# Level raising by prime powers

Suppose $f$ is a weight $2$ level $N$ cusp form. When can we realize the mod-$\ell$ representation of $f$ in a form of weight $2$ and level $Np^3$, where $p$ is some prime not dividing $N$? I assume that, if a simple criterion exists at all, it is a condition on the mod-$\ell$ representation of $f$ restricted to inertia at $p$, but I'm not sure what it would say...

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