Subconvexity for twisted $L$-functions on ${\mathrm{G}\mathrm{L}}_3$ over the Gaussian number field

Let $q\in \mathbb{Z}[i]$ be prime and $\chi$ be the primitive quadratic Hecke character modulo $q$. Let $\pi$ be a self-dual Hecke automorphic cusp form for ${\mathrm{S}\mathrm{L}}_3(\mathbb{Z}[i])$ and $f$ be a Hecke cusp form for ${\mathrm{\Gamma }}_0(q)\subset {\mathrm{S}\mathrm{L}}_2(\mathbb{Z}[i])$. Consider the twisted $L$-functions $L(s,\pi \otimes f\otimes \chi )$ and $L (s, \pi \otimes \chi)$on$$\mathrm{G}\mathrm{L}$_3 \times $\mathrm{G}\mathrm{L}$2$and$$\mathrm{G}\mathrm{L}$3$. We prove the subconvexity bounds \begin{equation*} L \big(\tfrac 1 2, \pi \otimes f \otimes \chi \big) \ll{, \varepsilon, \pi, f } $\mathrm{N}$ (q)^{5/4 + \varepsilon}, L \big(\tfrac 1 2 + it, \pi \otimes \chi \big) \ll{, \varepsilon, \pi, t } $\mathrm{N}$ (q)^{5/8 + \varepsilon}, \end{equation*} for any $\epsilon >0$.