Let $f$ be an eigenform of level $\Gamma_1(N)$ and let $p$ be a prime that does not divide $N$. It is well know that there is a $2$-dimensional representation $\rho_f$ of the absolute Galois group of $\mathbb Q$ attached to $f$. We can restrict $\rho$ to a decomposition subgroup at $p$ (after having fixed the usual isomorphisms), obtaining a representation $\rho_{f,p}$ of the absolute Galois group of $\mathbb Q_p$.

Is it possible to give an explicit description of this representation? Note that I'm in the case $p$ does not divide $N$, that should be easier than the case $p | N$.

Let $f$ be an eigenform of level $\Gamma_1(N)$ and let $p$ be a prime that does not divide $N$. It is well know that there is a $2$-dimensional representation $$ \rho_f \colon G_{\mathbb Q} \to \operatorname{GL}_2(E), $$ where $E$ is a finite extension of $\mathbb Q_p$, of the absolute Galois group of $\mathbb Q$ attached to $f$. We can restrict $\rho$ to a decomposition subgroup at $p$ (after having fixed the usual isomorphisms), obtaining a representation $\rho_{f,p}$ of the absolute Galois group of $\mathbb Q_p$.

Is it possible to give an explicit description of this representation? Note that I'm in the case $p$ does not divide $N$, that should be easier than the case $p | N$.