Hybrid bounds on twisted L-functions associated with modular forms

For f a primitive holomorphic cusp form of even weight k≥4, level N, and χ a Dirichlet character mod Q with (Q,N)=1, we establish the following subconvex hybrid bound for t∈R,L(12+it,fχ)≪Q38+θ4+ε(1+|t|)13−2θ+ε, where θ is the best bound toward the Ramanujan-Petersson conjecture at the infinite place. The implied constant only depends on f and ε. This is done via amplification and taking advantage of a shifted convolution sum of two variables as defined and analyzed in [9].