2015_7 Complex Indeed! -part 2

Geometry Level 4

\[\sum _{ m=1 }^{ 4n+1 }{ { \left\{ \sum _{ k=1 }^{ m-1 }{ \left\{ \sin\left( \frac { 2k\pi }{ m } \right) -i\cos\left( \frac { 2k\pi }{ m } \right) \right\} } \right\} }^{ m } } \]

where \(i=\sqrt { -1 } \).

What can we describe for the expression above?

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