Supercuspidal ramifications and traces of adjoint lifts at good primes

In this paper, we write down the local Brauer classes of the endomorphism algebras of motives attached to non-CM primitive Hecke eigenforms for all supercuspidal primes in terms of traces of adjoint lifts at auxiliary primes. We give an alternative proof of the result for odd primes $p$ obtained in [MR3391026] and write down the ramification formulas for odd unramified supercuspidal primes of level zero also removing a mild hypothesis of [MR3391026]. We also give a complete description of ramifications for $p=2$. The philosophy of adjoint lifts help us to determine the local Brauer classes at non dihedral primes by using results similar to [MR2770587]. We provide some numerical examples using {Sage} and {LMFDB} supporting some of our theorems.