Square orbits of additive characters of a finite field

The diagonal subgroup of $S{L}_2({\mathbb{F}}_p)$ conjugates $u(x)$ to $u(a^{2}x)$. It therefore has two orbits, each of size $(p-1)/2$, on the nontrivial characters $\chi (v\cdot )$ of the unipotent subgroup: one indexed by squares and one by nonsquares.