What is the relation between level and conductor of a supercuspidal representation of $\operatorname{GL}\_2(\mathbb{Q}\_p)$ for some prime $p$?


Proposition 3.4 in [Loeffler and Weinstein - On the computation of local components of a newform](https://arxiv.org/abs/1008.2796) refers to [Breuil and Mézard - Multiplicités modulaires et représentations de $\operatorname{GL}\_2(\mathbb Z\_p)$
et de $\operatorname{Gal}(\overline{\mathbb Q}\_p/\mathbb Q\_p)$ en $\ell = p$. Appendice par Guy Henniart. Sur l'unicité des types pour $\operatorname{GL}\_2$](https://www.imo.universite-paris-saclay.fr/~breuil/PUBLICATIONS/multiplicite.pdf), which I am unable to understand because of the language.



Level vs. conductor of a supercuspidal representation
There's probably a more elementary reference, but, according to [Bushnell, Henniart, and Kutzko - Local Rankin–Selberg convolutions for $\operatorname{GL}\_n$](https://www.ams.org/journals/jams/1998-11-03/S0894-0347-98-00270-7), (6.1.2), if $m$ is the level of $\pi$, then the conductor of $\pi$ depends on a choice of additive character $\psi$, which will be trivial on $\mathfrak p^{c(\psi)}$ but not on $\mathfrak p^{c(\psi) - 1}$ for some integer $c(\psi)$, and is given by
$$
f(\pi) = 2(1 + c(\psi) + m/e),
$$
where $e$ is $1$ if $\pi$ is unramified and $2$ if $\pi$ is ramified.


EDIT: I'll leave this answer since it's been accepted, but @Kimball's [comment](https://mathoverflow.net/questions/360614/level-vs-conductor-of-a-supercuspidal-representation#comment908588_360614) provides a better, as more elementary, reference in Section 2.2 of his paper [Kimball - The basis problem revisited](https://arxiv.org/abs/1804.04234).


