Steinberg representation of GL2 over a finite field ($\mathrm{St}$)

The action of $G{L}_2({\mathbb{F}}_q)$ on ${\mathbb{P}}^1({\mathbb{F}}_q)$ has permutation representation $1\oplus \mathrm{St}$. The Steinberg constituent is irreducible of dimension $q$ and has character values $q,0,1,-1$ on scalar, nontrivial Jordan, split regular semisimple, and nonsplit semisimple elements respectively.