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  "body": "\n## Definition\n\nLet $k$ be an integer and let $f$ be a [[modular form|GWKN5VG78A]] of weight $k$ for the [[modular group|VEKV1EM77B]] $\\mathrm{SL}_2(\\Z)$, with $q$-expansion $f(\\tau) = \\sum_{n \\ge 0} a_n q^n$, where $q = e^{2 \\pi i \\tau}$.\nThen $f$ is a *cusp form* if $a_0 = 0$.\nEquivalently, $f(\\tau) \\to 0$ as $\\operatorname{Im} \\tau \\to \\infty$.\n\n## Remarks\n\nFor a [[congruence subgroup|1H6R856P3C]] $\\Gamma$, a modular form $f$ for $\\Gamma$ of weight $k$ is a cusp form if $(c\\tau + d)^{-k} f(\\gamma \\tau) \\to 0$ as $\\operatorname{Im} \\tau \\to \\infty$, for every $\\gamma = \\begin{pmatrix} a \u0026 b \\\\ c \u0026 d \\end{pmatrix} \\in \\mathrm{SL}_2(\\Z)$.\nExample: $\\Delta = (E_4^3 - E_6^2) / 1728$ is a cusp form of weight $12$; the constant terms of $E_4^3$ and $E_6^2$ are both $1$, so they cancel.\nNon-example: the [[Eisenstein series|EYREVY31BN]] $E_4 = 1 + 240 q + \\cdots$ is a modular form but not a cusp form, since its constant term is $1$.\nFor $\\mathrm{SL}_2(\\Z)$, $S_k = 0$ for $k \u003c 12$ and $S_{12} = \\C \\Delta$.\n",
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    "name": "Cusp form",
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  "markdown": "https://math2.tamnd.com/node/SV5EEBF861.md",
  "msc": [
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  "name": "Cusp form",
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