This question is a variant of [this](https://mathoverflow.net/questions/173500) one.


Let $f$ be as in the other question, but suppose that we look at the $\ell$-adic representiation attached to $f$:
$$
\rho\_f : G\_{\mathbb Q} \to \operatorname{GL}\_2(E),
$$
where $E$ is a finite extension of $\mathbb Q\_\ell$, with $\ell$ a prime that does not divide $N$. Let $p \neq \ell$ be another prime that does not divide $N$.


What can be said about $\rho\_{f,p}$, the restriction of $\rho\_f$ to a decomposition subgroup at $p$?



About the restriction of a modular representation to a decomposition subgroup II
The following is (very) well known. Let's assume $f$ is a normalised newform of weight $k \geq 2$, nebentype $\chi$ with $p$th Fourier coefficient $a\_p$. Then your representation $\rho\_{f,p}$ (obtained as a piece of the cohomology of the modular curve) is unramified, and the characteristic polynomial of Frobenius is $X^2 - a\_p X + \chi(p)p^{k-1}$. This is the Eichler-Shimura relation, proved in this generality following the argument in Deligne's original paper.


In weight $k = 1$, the Galois representation has finite image and so Frobenius is always semisimple. In weight $k \ge 2$, it is conjectured that the polynomial $X^2 - a\_p X + \chi(p)p^{k-1}$ always has distinct roots, and hence Frobenius is always expected to be semisimple. This is known in weight $k = 2$, but unknown in general, although it does follow from the Tate conjecture; see <http://www.wstein.org/people/coleman/papers/coleman-edixhoven.pdf>


