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  "body": "\n## Definition\n\nLet $k$ be an integer, let $\\mathbb{H}$ be the [[upper half-plane|X6XGGH5FPW]], and let $M_k$ be the space of [[modular forms|GWKN5VG78A]] of weight $k$ for $\\mathrm{SL}_2(\\Z)$.\nFor an integer $n \\ge 1$, the *Hecke operator* $T_n \\colon M_k \\to M_k$ is given by\n$$(T_n f)(\\tau) = n^{k-1} \\sum_{\\substack{ad = n \\\\ a, d \\ge 1}} \\ \\sum_{b = 0}^{d - 1} d^{-k} f\\left(\\frac{a\\tau + b}{d}\\right).$$\n\n## Remarks\n\nIf $f = \\sum_{m \\ge 0} c(m) q^m$ with $q = e^{2 \\pi i \\tau}$, then $T_n f = \\sum_{m \\ge 0} \\gamma(m) q^m$ with\n$$\\gamma(m) = \\sum_{d \\mid \\gcd(m, n)} d^{k-1} c(mn/d^2).$$\nThe operators commute, $T_m T_n = T_{mn}$ when $\\gcd(m, n) = 1$, and $T_p T_{p^r} = T_{p^{r+1}} + p^{k-1} T_{p^{r-1}}$ for primes $p$ and $r \\ge 1$.\nThey map the [[cusp forms|A50Y012RBP]] $S_k$ into $S_k$.\nExample: $S_{12} = \\C \\Delta$ is one-dimensional, so $T_n \\Delta$ is a multiple of $\\Delta$; comparing the coefficients of $q$ gives $T_n \\Delta = \\tau(n) \\Delta$.\nNon-example: the map $f(\\tau) \\mapsto f(n\\tau)$ alone does not preserve $M_k$.\nFor instance, $E_4(2\\tau) = 1 + 240 q^2 + \\cdots$ is not a multiple of $E_4 = 1 + 240 q + \\cdots$, while $M_4 = \\C E_4$.\n",
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    "name": "Hecke operator",
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