Abstract: Serre's conjecture has two parts: a modularity part — that predicts which mod-ell Galois representations come from a modular form — and a minimal-lifting part — that predicts the minimal weight, level, and character of the modular form. Not long after formulating the conjecture, Serre found a counterexample to the minimal-lifting part for small values of ell (ell<5), where the character part of the prediction is wrong. In this talk, I'll discuss a variant of the minimal-lifting conjecture where the nebentypus character is replaced by the character used to define a supercuspidal representation. I will show that, in this case, even for ell=3, there is no obstruction to finding minimal lifts. This is joint work with Patrick Allen.