#!/usr/bin/env python
# coding: utf-8

# \begin{align}
#     \mathbf{C}_k\cdot \mathbf{C}_l & = \sum_{n=1}^N \cos\left(\frac{2\pi k n}{N} \right) \cos\left(\frac{2\pi l n}{N} \right)\\
#     &= \sum_{n=1}^N \frac{1}{2} \left(
#         \cos\left(\frac{2\pi n}{N} (k+l)\right) + 
#         \cos\left(\frac{2\pi n}{N} (k-l)\right) 
#         \right)\\
#     &= \frac{N}{2}\, \delta(k-l)  \\
#      \mathbf{S}_k\cdot \mathbf{S}_l &= \frac{N}{2}\,\delta(k-l)\\
#     \mathbf{C}_k\cdot \mathbf{S}_l &= 0\\
# \end{align}

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