Coupling a few 1008 roots!

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 $n =504$ .