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Let q be a prime and -D < -4 be an odd fundamental discriminant such that qsplits in Q(\sqrt{-D}). For f a weight zero Hecke-Maass newform of level q andh the weight one theta series of level D corresponding to an ideal class groupcharacter of Q(\sqrt{-D}), we establish a hybrid subconvexity bound for L(f\times h,s) at the central point when q = D^{\eta} for 0 < \eta < 1. With thiscircle of ideas, we show that the Heegner points of level q and discriminant Dbecome equidistributed, in a natural sense, as q, D become large with q <D^{1/20-\varepsilon}. Our approach to these problems is connected to estimatingthe L^2-restriction norm of a Maass form of large level when restricted to thecollection of Heegner points. We furthermore establish bounds for quadratictwists of Hecke-Maass L-functions with simultaneously large level and largequadratic twist, and hybrid bounds for quadratic Dirichlet L-functions incertain ranges.

Subconvexity and equidistribution of Heegner points in the level aspect