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  "body": "\n## Definition\n\nLet $k$ be an integer, and let $\\mathbb{H}$ be the [[upper half-plane|X6XGGH5FPW]] with coordinates $\\tau = x + iy$.\nLet\n$$\\mathcal{F} = \\set{\\tau \\in \\mathbb{H} : \\abs{\\tau} \\ge 1, \\ \\abs{x} \\le 1/2}$$\nbe the standard fundamental domain of the [[modular group|VEKV1EM77B]].\nLet $f$ and $g$ be [[modular forms|GWKN5VG78A]] of weight $k$ for $\\mathrm{SL}_2(\\Z)$.\nAssume that $f$ or $g$ is a [[cusp form|A50Y012RBP]].\nTheir *Petersson inner product* is\n$$\\langle f, g \\rangle = \\int_{\\mathcal{F}} f(\\tau)\\, \\overline{g(\\tau)}\\, y^k \\, \\frac{dx\\, dy}{y^2}.$$\n\n## Remarks\n\nThe function $f \\overline{g} y^k$ and the measure $dx\\, dy / y^2$ are invariant under $\\mathrm{SL}_2(\\Z)$, so any fundamental domain gives the same value.\nA cusp form decays like $e^{-2\\pi y}$ as $y \\to \\infty$, so the integral converges.\nOn the space $S_k$ of cusp forms, $\\langle \\cdot, \\cdot \\rangle$ is a positive definite Hermitian [[inner product|0NB0AWGJBW]].\nExample: $\\langle \\Delta, \\Delta \\rangle \u003e 0$, since $\\Delta$ has no zeros on $\\mathbb{H}$, so the integrand $\\abs{\\Delta}^2 y^{12} \\cdot y^{-2}$ is positive on $\\mathcal{F}$.\nNon-example: for $f = g = E_4$ the integral diverges, because $\\abs{E_4}^2 y^4 \\cdot y^{-2}$ tends to infinity like $y^2$ as $y \\to \\infty$; this is why one form must be a cusp form.\nFor a congruence subgroup $\\Gamma$, the integral is taken over a fundamental domain of $\\Gamma$, often divided by the index of $\\Gamma$ in $\\mathrm{SL}_2(\\Z)$.\n",
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