Stiefel-Whitney Classes for Finite Symplectic Groups

Let $q$ be an odd prime power, and $G=\text{Sp}(2n,q)$ the finite symplectic group. We give an expression for the total Stiefel-Whitney Classes (SWCs) for orthogonal representations $\pi$ of $G$, in terms of character values of $\pi$ at elements of order $2$. We give "universal formulas’’ for the fourth and eighth SWCs. For $n=2$, we compute the subring of the mod $2$ cohomology generated by the SWCs $w_k(\pi )$.