A generalized cubic moment and the Petersson formula for newforms

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.

Explicit calculations with Eisenstein series

“…Returning to Theorem 7.1, we see that Taking B = 1, we get 1 4π E t , E t = ν(N), and so (10.7) follows, as well as (10.8) E t , E t N = ν(N) E t , E t 1 where the subscript on the inner product symbol denotes the level of the group to which the inner product is attached. Hence where λ E (n) = τ iT (n); this is the analog of [PY,(2.8)]. We therefore need to evaluate the inner sum over φ, namely (10.9)…”

“…Here (10.9) is analogous to [PY,(3.2)]. At this point, all the calculations of T t (m, n) run completely parallel to those in [PY,Section 3], since the formulas that were used there are: Hecke relations, (10.7), and (10.8), which are the same in both cases of cusp forms vs. Eisenstein. Therefore, (10.4) holds with x = ∞.…”

Moments of Spinor L-Functions and Symplectic Kloosterman Sums

“…cf. [7, §2.5], and λpφq, µpφq are given with respect to (14) by λpφq " xd 1 e 1 e 0 e 1 φ, φy xφ, φy , µpφq " xd 1 e 0 e 1 e 0 φ, φy xφ, φy .…”

“…In the GLp2q case, such a formula is well-known and derived by first constructing an explicit orthogonal basis of oldforms and then applying Möbius inversion to sieve these forms out, cf. [14].…”

Mixed moment of GL(2) and GL(3)L‐functions

“…Another example is due to Petrow who refined the estimate of Conrey and Iwaniec for the cubic moment of central ‐values of level cusp forms twisted by quadratic characters of conductor and showed that the role of the dual moment is played by the weighted fourth moment of Dirichlet ‐functions [, Theorems 1,2]. See also for related recent results.…”