How would one go about computing the time complexity of a Jacobi sum, such as $$J_5(\chi,\chi)=\sum_{s=0}^{p-1}\chi(s)\chi(1-s),$$ where $\chi$ is a character of $\mathbb{F}_p$ mapping a fixed non-quintic element in $\mathbb{F}_p$ to $\zeta=e^{2\pi i/5}$ and $p$ is a prime congruent to 1 mod 5?
In particular, if we were to write $$J_5(\chi,\chi)=c_0+c_1\zeta+c_2\zeta^2+c_3\zeta^3+c_4\zeta^4,$$
what would be the big-O for finding the $c_i$?