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  "body": "\n## Definition\n\nLet $K$ be a [[number field|MWBDFS2KJ5]], let $I_K$ be its idele group and let $C_K = I_K / K^\\times$ be its [[idele class group|9GHD9MXNYV]].\nA *Hecke character* of $K$ is a continuous [[group homomorphism|0JYTG5K9Y4]]\n$$\\chi \\colon I_K \\to \\C^\\times$$\nsuch that $\\chi(a) = 1$ for every $a \\in K^\\times$.\nEquivalently, it is a continuous homomorphism $C_K \\to \\C^\\times$.\n\n## Remarks\n\nExample: the idele norm $\\abs{x} = \\prod_v \\abs{x_v}_v$ is continuous and multiplicative, and $\\abs{a} = 1$ for $a \\in K^\\times$ by the product formula; so it is a Hecke character, and so is $\\abs{\\cdot}^s$ for every $s \\in \\C$.\nNon-example: for $K = \\Q$, the map $x \\mapsto \\abs{x_\\infty}$ that keeps only the real component is a continuous homomorphism, but it sends the rational number $2$ to $2 \\neq 1$.\nSome authors call $\\chi$ a Hecke character only when $\\abs{\\chi(x)} = 1$ for all $x$, and call the general case a quasi-character or Größencharakter.\nEvery Hecke character is a unitary one times $\\abs{\\cdot}^\\sigma$ for a real number $\\sigma$.\nThe Hecke characters of $\\Q$ of finite order correspond to the primitive [[Dirichlet characters|7XB5VJKDPG]].\n",
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