Modular analogs of character formulas and minimal lifts of modular forms

If $f$ is a mod-$3$ eigenform of weight 2 and level $\Gamma_0(\ell^2)$ for a prime $\ell$ such that $\ell \equiv -1 \pmod{3}$, and $\ell$ is a vexing prime for $f$, we show that there is no obstruction to finding a minimal lift of $f$, but that there is an obstruction to finding a nonminimal lift. The key new ingredient that we prove is a modular analog of the standard character formula for a cuspidal representation of $\mathrm{GL}_2(\mathbb{F}_\ell)$, an enhancement that allows us to easily compute the group cohomology of a $3$-adic lattice in such a representation. In fact, we provide a general framework for proving such modular analogs for a broader class of representations using results of Brou\'e and Puig in modular representation theory. We show that this class includes certain Deligne--Lusztig representations and representations coming from higher-depth supercuspidal representations of $\mathrm{GL}_2$.