The cubic moment of $L$-functions for specified local component families

The construction and estimation of appropriate cubic moments of $L$-functions in small families
has been responsible for the strongest-known bounds for degree $2$ automorphic $L$-functions.
The Weyl bound for all Dirichlet $L$-functions is a consequence of this line of work. In previous
work with Petrow, we constructed families of twisted $L$-functions which may be interpreted as
capturing the $L$-functions whose underlying automorphic representation is everywhere principal
series. I will discuss recent work with Hu and Petrow that extends these methods to also treat
supercuspidal representations. In particular, we obtain the Weyl bound for level $p^2$ cusp forms
which are twist-minimal. The method relies on new versions of the Petersson/Bruggeman/Kuznetsov
formula and on the fourth moment of Dirichlet $L$-functions along a coset.