A generalized cubic moment and the Petersson formula for newforms

TL;DR

This paper establishes a subconvexity bound for quadratic twists of holomorphic newforms with square-free level using a novel cubic moment approach and a generalized Petersson formula, extending previous results to broader cases.

Contribution

It introduces a generalized Petersson formula for square-free level newforms and applies a cubic moment method to achieve a Weyl-type subconvexity bound for quadratic twists.

Findings

Proved a Weyl-type subconvexity bound for quadratic twists of newforms.

Developed a more general Petersson formula for square-free level newforms.

Extended previous work to newforms with arbitrary square-free level and quadratic characters with even conductor.

Abstract

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.