Stiefel-Whitney Classes of Representations of $\text{SL}(2,q)$

We describe the Stiefel-Whitney classes (SWCs) of orthogonal representations $\pi$ of the finite special linear groups $G=\text{SL}(2,\mathbb F_q)$, in terms of character values of $\pi$. From this calculation, we can answer interesting questions about SWCs of $\pi$. For instance, we determine the subalgebra of $H^*(G,\mathbb Z/2\mathbb Z)$ generated by the SWCs of orthogonal $\pi$, and we also determine which $\pi$ have nontrivial mod $2$ Euler class.