Modular analogs of character formulas and minimal lifts of modular forms

If $f$ is a mod-$3$ eigenform of weight 2 and level ${\mathrm{\Gamma }}_0({\mathrm{\ell }}^2)$ for a prime $\mathrm{\ell }$ such that $\mathrm{\ell }\equiv -1(\mathrm{m}\mathrm{o}\mathrm{d}3)$, and $\mathrm{\ell }$ 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 ${\mathrm{G}\mathrm{L}}_2({\mathbb{F}}_{\mathrm{\ell }})$, 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 Broué 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 ${\mathrm{G}\mathrm{L}}_2$.