Small $\text{PSL}(2,\mathbb{F})$ representations of Seifert fiber space groups

Let $M$ be a Seifert fiber space with non-abelian fundamental group and admitting a triangulation with $t$ tetrahedra. We show that there is a non-abelian $\text{PSL}(2,\mathbb{F})$ quotient where $\mathrm{\mid }\mathbb{F}\mathrm{\mid }<c(2^{20t}{3}^{120t})$ for an absolute constant $c>0$ and use this to show that the lens space recognition problem lies in coNP for Seifert fiber space input. We end with a discussion of our results in the context of distinguishing lens spaces from other $3$–manifolds more generally.