A subconvex bound for twisted $L$-functions

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

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