Let $w \in G_{24}$ be a primitive root. Find every $n \in \mathbb{N}$ such that $w^{15n} + \sum^{11}_{i=1} \overline{w}^{2i+6}$ is purely imaginary.

Question

We know that $\overline{w} = w^{23}$. Given that $(24:23)=1$, $w^{23}$ is also primitive. But, for any such root $u$, $u^2 \in G_{12}$ is another primitive root. Hence,

$$\sum^{11}_{i=1} \overline{w}^{2i+6}= \overline{w}^6\sum^{11}_{i=1} (\overline{w}^{2})^i = w^{18}\bigg(\sum^{11}_{i=0} (u^{2})^i - 1 \bigg) = -w^{18}$$

So we're looking for numbers that look like this:

$$w^{15n} - w^{18}$$

...and that have no real part. Am I too off the mark? How can I finish this exercise?