# Coupling a few 1008 roots!

Algebra Level 5

Consider the $$(2n)^\text{th}$$ roots of unity: $$1, \omega, \omega^2, \ldots, \omega^{2n-1}$$.

Let $$\zeta$$ be a complex number with value of $$\omega\omega^{2n} + \omega^2 \omega^{2n-1} + \cdots + \omega^{2n} \omega$$.

Let $$\xi$$ denotes the absolute value of $$\zeta$$, and $$\alpha$$ denotes its argument, find $$m = \dfrac{a\xi}{\pi}$$ when $$n =504$$.

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