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

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

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