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Using a cubic moment, we prove a Weyl-type subconvexity bound for thequadratic twists of a holomorphic newform of square-free level, trivialnebentypus, and arbitrary even weight. This generalizes work of Conrey andIwaniec in that the newform that is being twisted may have arbitrarysquare-free level, and also that the quadratic character may have evenconductor. One of the new tools developed in this paper is a more generalPetersson formula for newforms of square-free level.

A generalized cubic moment and the Petersson formula for newforms