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