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  "body": "\n## Summary\n\nA cusp form is a [[modular form|GWKN5VG78A]] that vanishes at every cusp.\nFor the full [[modular group|VEKV1EM77B]], this means that the constant term of its $q$-expansion is $0$.\nThe cusp forms of weight $k$ form a subspace $S_k$ of $M_k$, on which the [[Petersson inner product|EMZTGD63V3]] is defined.\nThe coefficients of cusp forms grow more slowly than those of Eisenstein series, and they carry deep arithmetic information.\n\n## Explanations\n\n### L1 human\n\nA modular form has an expansion $a_0 + a_1 q + a_2 q^2 + \\cdots$ with $q = e^{2 \\pi i \\tau}$.\nAs $\\tau$ goes up to infinity, $q$ goes to $0$, and the value tends to $a_0$.\nA cusp form is one with $a_0 = 0$, so it dies off at infinity.\nThe first example is the discriminant $\\Delta = q \\prod_{n \\ge 1} (1 - q^n)^{24}$ of weight $12$.\nIts coefficients are the values of the [[Ramanujan tau function|6DNZAGCSQS]].\n",
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