A bound for twists of $\rm GL_3\times GL_2$ $L$-functions with composite modulus

Let $π$ be a Hecke-Maass cusp form for $\rm SL_3(\mathbf{Z})$ and let $g$ be a holomorphic or Maass cusp form for $\rm SL_2(\mathbf{Z})$. Let $χ$ be a primitive Dirichlet character of modulus $M=M_1M_2$ with $M_i$ prime, $i=1,2$. Suppose that $M^{1/2+2η}<M_1<M^{1-2η}$ with $0<η<1/8$. Then we have $L\left(\frac{1}{2},π\otimes g \otimes χ\right)\ll_{π,g,\varepsilon} M^{3/2-η+\varepsilon}.$

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