How prove this $\sum_{k=0}^{n-1}\left|\sum_{j=-n}^{n-1}p_{j}e^{ikj\pi/n}\right|^2=2n\sum_{j=-n}^{n-1}|p_{j}|^2$

Question

show that $$\displaystyle\sum_{k=0}^{n-1}\left|\sum_{j=-n}^{n-1}p_{j}e^{ikj\pi/n}\right|^2=2n\displaystyle\sum_{j=-n}^{n-1}|p_{j}|^2=\dfrac{n}{\pi}||\displaystyle\sum_{j=-n}^{n-1}p_{j}e^{ijt}||_{L^2}^{2}$$

where $i=\sqrt{-1}$

Thank you everyone