Abstract We develop generalized Petersson/Bruggeman-Kuznetsov (PBK) formulas for specified local components at non-archimedean places. In fact, we introduce two hypotheses on non-archimedean test function pairs f (f), called geometric and spectral hypotheses, under which one obtains “nice” PBK formulas by the adelic relative trace function approach. Then, given a supercuspidal representation of {PGL}₂ (Qₚ), we study extensively the case that (f) is a projection onto the line of the newform if is isomorphic to or its unramified quadratic twist, and (f) = 0 otherwise. As a first application, we prove an optimal large sieve inequality for families of automorphic representations that arise in our framework.
Hu et al. (Thu,) studied this question.