Nonvanishing of special $L$-values of cusp forms on $\mathrm{G}\mathrm{L}(2)$ with totally split prime power twists: without the class number hypothesis

Let $F$ be a number field and $f$ a normalized cuspidal Hecke eigenform on $\mathrm{G}\mathrm{L}(2)$ over $F$, with $k\mathrm{/}2$ as the central critical point of its $L$-function $L(f,s)$. Let $\mathfrak{p}$ be a totally split prime ideal of $F$ lying above an odd prime $p$ and coprime to the level ideal of $f$. In this article, we show that $L(f\otimes \phi ,k\mathrm{/}2)$ does not vanish for sufficiently ramified Hecke characters of $F$ of $p$-power order and $\mathfrak{p}$-power conductor. Our result improves previous works of Jaesung Kwon and Hae-Sang Sun, in which the additional hypothesis that $p$ does not divide the class number of $F$ is required.