Let $w\in G_{18}$ be a primitive root of unity. Prove that $w^{16} \sum_{j=1}^{17}(w^3 \overline{w})^{3j+1}$ is imaginary pure.

Question

This is what I've got:

$$w^3\overline{w}=w^3w^{-1}=w^2 \iff \\ w^{16} \sum_{j=1}^{17}(w^3 \overline{w})^{3j+1} = w^{16}w^2\sum_{j=1}^{17}w^{6j} = \bigg( \sum_{j=0}^{17}w^{6j} \bigg) - 1$$

Given that $18 = 6\times 3$, we know that $w^6 \in G_3$ is another primitive root of unity.

How can I move on from here?