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).