This was an old homework problem that I wasn't able to solve and seem to have lost the solution to. If $p$ is a prime number greater than $3$, then the legendre symbol $(\frac{a}{p})$ as a function of $a$ is the unique Dirichlet character of order $2$. That's easy enough to show, but I've been unable to figure out the next part.
The problem is to show that, if $\chi$ is any nontrivial character on $\mathbb{Z}_p$, then $\sum\limits_{t \in \mathbb{Z}_p}\chi(1-t^2) = \sum\limits_{a+b=1}\chi(a)(\frac{b}{p})$, where $a, b \in \mathbb{Z}_p$.