Bounds for $GL_3$ $L$-functions in depth aspect

Let $f$ be a Hecke-Maass cusp form for $SL_3(\mathbb{Z})$ and $\chi$ a primitive Dirichlet character of prime power conductor $\mathfrak{q}=p^{\kappa}$ with $p$ prime and $\kappa\geq 10$. We prove a subconvexity bound $$L\left(\frac{1}{2},\pi\otimes \chi\right)\ll_{p,\pi,\varepsilon} \mathfrak{q}^{3/4-3/40+\varepsilon}$$ for any $\varepsilon>0$, where the dependence of the implied constant on $p$ is explicit and polynomial. We obtain this result by applying the circle method of Kloosterman's version, summation formulas of Poisson and Voronoi's type and a conductor lowering mechanism introduced by Munshi [14]. The main new technical estimates are the essentially square root bounds for some twisted multi-dimensional character sums, which are proved by an elementary method.