$$\because$$ $${z^2} + z + 1 = 0$$

$$\Rightarrow$$ $$\omega$$ or $$\omega$$²

$$\because$$ $$\left| {\sum\limits_{n = 1}^{15} {{{\left( {{z^n} + {{( - 1)}^n}{1 \over {{z^n}}}} \right)}^2}} } \right|$$

$$ = \left| {\sum\limits_{n = 1}^{15} {{z^{2n}} + \sum\limits_{n = 1}^{15} {{z^{ - 2n}} + 2\,.\,\sum\limits_{n = 1}^{15} {{{( - 1)}^n}} } } } \right|$$

$$ = \left| {0 + 0 - 2} \right|$$

$$ = 2$$