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  "body": "\n## Definition\n\nLet $k$ be an integer and let $\\mathbb{H} = \\set{\\tau \\in \\C : \\operatorname{Im} \\tau \u003e 0}$ be the [[upper half-plane|X6XGGH5FPW]].\nA *modular form of weight $k$* for the [[modular group|VEKV1EM77B]] $\\mathrm{SL}_2(\\Z)$ is a function $f \\colon \\mathbb{H} \\to \\C$ such that:\n\n1. $f$ is [[holomorphic|0NAHXR1FRH]] on $\\mathbb{H}$;\n2. $f\\left(\\frac{a\\tau + b}{c\\tau + d}\\right) = (c\\tau + d)^k f(\\tau)$ for all $\\tau \\in \\mathbb{H}$ and all $\\begin{pmatrix} a \u0026 b \\\\ c \u0026 d \\end{pmatrix} \\in \\mathrm{SL}_2(\\Z)$;\n3. $f$ is bounded on the region $\\set{\\tau \\in \\mathbb{H} : \\operatorname{Im} \\tau \\ge 1}$.\n\n## Remarks\n\nCondition 2 with $\\begin{pmatrix} 1 \u0026 1 \\\\ 0 \u0026 1 \\end{pmatrix}$ gives $f(\\tau + 1) = f(\\tau)$, so $f(\\tau) = \\sum_{n \\in \\Z} a_n q^n$ with $q = e^{2 \\pi i \\tau}$.\nCondition 3 then says $a_n = 0$ for $n \u003c 0$; this is \"holomorphic at infinity\".\nExample: every constant function is a modular form of weight $0$.\nExample: the [[Eisenstein series|EYREVY31BN]] $G_4$ is a nonzero modular form of weight $4$.\nNon-example: the [[j-invariant|FEC2XH822R]] satisfies 1 and 2 with $k = 0$, but $j = q^{-1} + 744 + \\cdots$ is unbounded as $\\operatorname{Im} \\tau \\to \\infty$, so 3 fails.\nIf $k$ is odd, the matrix $-I$ gives $f = (-1)^k f$, so $f = 0$.\nFor a [[congruence subgroup|1H6R856P3C]] $\\Gamma$, condition 2 is required only for matrices in $\\Gamma$.\nCondition 3 is then required at every cusp: $(c\\tau + d)^{-k} f(\\gamma \\tau)$ is bounded on $\\operatorname{Im} \\tau \\ge 1$ for every $\\gamma \\in \\mathrm{SL}_2(\\Z)$.\nThe modular forms of weight $k$ form a finite-dimensional complex vector space $M_k$.\n",
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