Roots of Unity

Algebra Level 5

Let \(\omega\) be a complex number that is not a real number such that \(\omega^8 = 1\). Evaluate \[ \large \dfrac{1}{3} \sum_{k = 1}^{2016} \dfrac{\omega^{6k^{4}} + \omega^{5k^{4}} + \omega^{4k^{4}} + \omega^{3k^{4}} + \omega^{2k^{4}} + \omega^{k^{4}} + 1}{\omega^{6k^{4}} - \omega^{5k^{4}} + \omega^{4k^{4}} - \omega^{3k^{4}} + \omega^{2k^{4}} - \omega^{k^{4}} + 1} .\]

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