Square orbits of additive characters of a finite field
= Square orbits of additive characters of a finite field
The diagonal subgroup of $SL_2(\mathbb F_p)$ conjugates $u(x)$ to $u(a^2x)$. It therefore has two orbits, each of size $(p-1)/2$, on the nontrivial characters $\chi(v\mathbin\cdot)$ of the unipotent subgroup: one indexed by squares and one by nonsquares.