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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