Rationality of twists of the Siegel modular variety of genus 2 and level 3

Let $\overline {\rho }: G_{\mathbf {Q}} \rightarrow \operatorname {GSp}_4(\mathbf {F}_3)$ be a continuous Galois representation with cyclotomic similitude character. Equivalently, consider }$ to the associated $3$-torsion of principally polarized abelian surface $A/\mathbf {Q}$. We prove that moduli space $\mathcal {A}_2(\overline })$ surfaces $B/\mathbf {Q}$ admitting symplectic isomorphism $B[3] \simeq \overline representations is never rational over $\mathbf when surjective, even though it both {C}$ and unirational via map degree $6$.