Coupling a few 1008 roots!

Algebra Level 5

Consider the (2n)th(2n)^\text{th} roots of unity: 1,ω,ω2,,ω2n11, \omega, \omega^2, \ldots, \omega^{2n-1} .

Let ζ \zeta be a complex number with value of ωω2n+ω2ω2n1++ω2nω \omega\omega^{2n} + \omega^2 \omega^{2n-1} + \cdots + \omega^{2n} \omega .

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

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