Say $p\equiv 1\pmod n$ and $a$ is a primitive root$\pmod p$. Can we find or estimate the value of $\sum_{i=1}^{(p-1)/n} \zeta^{a^i}$ and its conjugates?
Note that if $n$ is odd, this sum is real by Galois theory (it is fixed by a subgroup of the Galois group of order $n$). Furthermore, its conjugates are of a similar form with $\zeta^{a^i}$ replaced with $\zeta^{t\cdot a^i}$.
As an example, if we take $7\equiv 1\pmod 3$, then we get $\zeta+\zeta^6$. In general, can we expect this sum to be small?