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