The circle method and bounds for $$L$$ L -functions - I

Abstract

Let \(f\) be a Hecke-Maass or holomorphic primitive cusp form of arbitrary level and nebentypus, and let \(\chi \) be a primitive character of conductor \(M\). For the twisted \(L\)-function \(L(s, f\otimes \chi )\) we establish the hybrid subconvex bound $$\begin{aligned} L\left( \frac{1}{2}+it, f\otimes \chi \right) \ll (M(3+|t|))^{\frac{1}{2}-\frac{1}{18}+\varepsilon }, \end{aligned}$$ for \(t\in \mathbb{R }\). The implied constant depends only on the form \(f\) and \(\varepsilon \).