In this talk, we describe some results about the local Galois representations attached to classical cusp forms, with the hope of applying them to some well-known problems.

We first describe the shape of the local mod $p$ representations attached to forms of weight $2$ or more, and small slope, concentrating on the case of slope one. The proof uses the mod $p$ Local Langlands correspondence. A solution for all slopes might shed some light on a generalization of Serre's uniform boundedness conjecture for elliptic curves to the case of modular abelian varieties / motives. This is joint work with Shalini Bhattacharya and Sandra Rozensztajn.

We also count the number of level preserving twists of the local representations attached to cusp forms of weight $1$ and minimal level. Such results might help in counting exotic weight 1 forms by reducing the problem to a count of number fields of small degree and by appealing to the work of Bhargava. This is ongoing joint work with Manjul Bhargava.