A subconvex bound for twisted $L$-functions

Abstract

Let $\mathfrak{q}>2$ be a prime number, $\chi$ a primitive Dirichletcharacter modulo $\mathfrak{q}$ and $f$ a~primitive holomorphic cusp form or a Hecke-Maass cusp formof level $\mathfrak{q}$and trivial nebentypus. We prove the subconvex bound $$L(1/2,f\otimes \chi)\ll \mathfrak{q}^{1/2-1/12+\varepsilon},$$ where the implicit constant depends only on the archimedean parameter of $f$ and $\varepsilon$. The main input is a modifyingtrivial delta methoddeveloped in [1].