The circle method and bounds for L-functions---II: Subconvexity for twists of GL(3) L-functions

Let $\pi$ be a ${\rm SL}(3,\Bbb{Z})$ Hecke-Maass cusp form. Let $\chi=\chi_1\chi_2$ be a Dirichlet character with $\chi_i$ primitive modulo $M_i$. Suppose $M_1$, $M_2$ are primes such that $\sqrt{M_2}M^{4\delta}<M_1<M_2M^{-3\delta}$, where $M=M_1M_2$ and $0<\delta<1/28$. In this paper we will prove the following subconvex bound $$ L\left({1\over 2},\pi\otimes\chi\right)\ll_{\pi,\varepsilon} M^{{3\over 4}-\delta+\varepsilon}. $$