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 of Carayol that the prime-to-$p$ conductor $N(\rho)$ of $\rho$ equals $N$. Hence, one can recover $N$ from $\{\rho\vert_{I_q}\}_{q\mid N}

The question is:

Can $r$ be read in $\rho\vert_{I_p}$?

Thanks for your time!