Let $z$ be a complex number such that $z^n = 1$ where $n = 6$. Let $\omega = e^{i \frac{2\pi}{6}}$ be a primitive 6th root of unity. Consider the set $S = \{1, \omega, \omega^2, \omega^3, \omega^4, \omega^5\}$. Which of the following values are equal to the sum $T = \sum_{k=0}^{5} (\omega^k)^2$?