Abstract

Let f(z)=a(m)m(k1)/2e(mz)Sk(N,χ1) be a primitive cusp form of integral weight k, level N, and character χ1. For a smooth weight function g with compact support in [M1,2M1]×[M2,2M2] and positive integers l1, l2, h, the bound l1m1l2m2=ha(m1)a(m2)¯g(m1,m2)1≪ε(l1M1+l2M2)1/2+θ+ε with θ = 7/64 is shown. As an application, the shifted sum mMa(m)a(m+h)¯ is bounded nontrivially for hM64/39ε. Furthermore, the subconvexity bound Lf(1/2+it,χ)εD71/167+ε for the L-function attached to the twist of f with a primitive character χ to modulus D is obtained.

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