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  "body": "\n## Summary\n\nA Dirichlet character modulo $q$ is a function $\\chi \\colon \\Z \\to \\C$ that comes from a homomorphism $(\\Z/q\\Z)^\\times \\to \\C^\\times$.\nIt is periodic with period $q$ and completely multiplicative.\nThere are exactly $\\varphi(q)$ characters modulo $q$, and their orthogonality relations isolate a single residue class.\nDirichlet introduced them to prove [[Dirichlet's theorem on arithmetic progressions|0GNVN3CA2P]], through the [[Dirichlet L-functions|9D2Z543EJQ]] $L(s, \\chi)$.\n\n## Explanations\n\n### L1 human\n\nA Dirichlet character tags each integer with a complex number that depends only on its remainder modulo $q$.\nIntegers that share a factor with $q$ get $0$, and the others get a [[root of unity|0YV2CKP2AV]].\nThe tags multiply: the tag of a product is the product of the tags.\nFor $q = 4$, the character gives $0$ to even numbers, $1$ to numbers $1 \\bmod 4$, and $-1$ to numbers $3 \\bmod 4$.\n",
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