\[\begin{array}{l} {z=-\frac{1}{2} +i\frac{\sqrt{3} }{2} ,\; \overline{z}=-\frac{1}{2} -i\frac{\sqrt{3} }{2} } \\ {|z|=|\overline{z}|=\sqrt{\left(\frac{-1}{2} \right)^{2} +\left(\frac{\sqrt{3} }{2} \right)^{2} } =1} \\ {\varphi (z)=\pi +arctg(-\sqrt{3} )=\frac{2\pi }{3} } \\ {z=1\cdot (\cos \frac{2\pi }{3} +i\sin \frac{2\pi }{3} )} \\ {\overline{z}=1\cdot (\cos \frac{2\pi }{3} -i\sin \frac{2\pi }{3} )} \\ {z^{2001} =\cos \frac{4002\pi }{3} +i\sin \frac{4002\pi }{3} } \\ {\overline{z}^{2001} =\cos \frac{4002\pi }{3} -i\sin \frac{4002\pi }{3} } \\ {z^{2001} =\cos (1334\pi )+i\sin (1334\pi )=1+i\cdot 0=1} \\ {\overline{z}^{2001} =\cos (1334\pi )-i\sin (1334\pi )=1-i\cdot 0=1} \\ {\left(-\frac{1}{2} +i\frac{\sqrt{3} }{2} \right)^{2001} +\left(-\frac{1}{2} -i\frac{\sqrt{3} }{2} \right)^{2001} = 2} \end{array}\]