Newforms in Cuspidal Representations

We consider newform vectors in cuspidal representations of $p$-adic general linear groups. We extend the theory from the complex setting to include~$\mathrm{\ell }$-modular representations with~$\mathrm{\ell }\mathrm{\ne }p$, and prove that the conductor is compatible with congruences modulo~$\mathrm{\ell }$ for (ramified) supercuspidal~$\mathrm{\ell }$-modular representations and for depth zero cuspidals. In the complex and modular setting, we prove explicit formulae for depth zero and minimax cuspidal representations of integral depth, in Bushnell-Kutzko and Whittaker models.