Hybrid bounds for $\mathrm{G}\mathrm{L}(4)\times \mathrm{G}\mathrm{L}(1)$ twisted $L$-functions

Let $P,M$ be a two primes such that $(P,M)=1$. Let $\mathrm{\Pi }$ be a normalized Hecke-Maaß form on $\mathrm{G}\mathrm{L}(4)$ of level $P$, and $\chi$ a primitive Dirichlet character modulo $M$. In this paper, we study the hybrid subconvexity problem for $L(s,\mathrm{\Pi }\otimes \chi )$ simultaneously in the level and conductor aspects. Among other things, we prove a hybrid subconvex bound, so long as $M^{1\mathrm{/}5}<P<M^{2\mathrm{/}5}$.