# On a hybrid bound for twisted *L*-values

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## Abstract

Let *f *∈*S* _{ k }(*M*, *ψ*) be a newform, and let χ be a primitive character of conductor *q*. We express \({L(\frac{1}{2}+it,f\otimes\chi)}\) as a short combination of bilinear forms involving Kloosterman fractions. Using this we establish the convexity breaking bound \({L\left(\tfrac{1}{2}+it,f\otimes\chi\right)\ll_{f,\varepsilon} [q(1+|t|)]^{\frac{1}{2}-\frac{1}{118}+\varepsilon}}\) for any *ε* > 0.

## Mathematics Subject Classification (2010)

Primary 11F66 Secondary 11M41## Keywords

*L*-functions Subconvexity Hybrid bound

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