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