  Let $\pi$ be a Hecke-Maass cusp form for $SL(3,\mathbb{Z})$ and
$\chi=\chi_1\chi_2$ a Dirichlet character with $\chi_i$ primitive modulo $M_i$.
Suppose that $M_1$, $M_2$ are primes such that
$\max\{(M|t|)^{1/3+2\delta/3},M^{2/5}|t|^{-9/20},
M^{1/2+2\delta}|t|^{-3/4+2\delta}\}(M|t|)^{\varepsilon}<M_1< \min\{
(M|t|)^{2/5},(M|t|)^{1/2-8\delta}\}(M|t|)^{-\varepsilon}$ for any
$\varepsilon>0$, where $M=M_1M_2$, $|t|\geq 1$ and $0<\delta< 1/52$. Then we
have $$ L\left(\frac{1}{2}+it,\pi\otimes \chi\right)\ll_{\pi,\varepsilon}
(M|t|)^{3/4-\delta+\varepsilon}. $$
