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In this paper we derive refined Petersson/Kuznetsov trace formulae withprescribed local ramifications. The spectral side of these formulae picks outnewforms whose associated local components come from specific sub-families ofrepresentations of given level, and are much shorter compared with theclassical versions. We use them to study the first moment and the subconvexitybound of certain Rankin-Selberg L-function in a hybrid setting, obtaining Weylbound in a wider range compared to previous works.

The Petersson/Kuznetsov trace formula with prescribed local ramifications