Siegel ${\mathfrak{p}}^2$ Vectors for Representations of $GSp(4)$

Let $F$ be a $p$-adic field and $(\pi ,V)$ an irreducible complex representation of $G=GSp(4,F)$ with trivial central character. Let $\mathrm{S}\mathrm{i}({\mathfrak{p}}^2)\subset G$ denote the Siegel congruence subgroup of level ${\mathfrak{p}}^2$ and $u\in N_G(\mathrm{S}\mathrm{i}({\mathfrak{p}}^2))$ the Atkin-Lehner element. We compute the dimension of the space of $\mathrm{S}\mathrm{i}({\mathfrak{p}}^2)$-fixed vectors in $V$ as well as the signatures of the involutions $\pi (u)$ acting on these spaces.