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  "body": "\n## Definition\n\nLet $N \\ge 1$ be an integer.\nThe *principal congruence subgroup of level $N$* is the [[kernel|0W88CEGQP1]] of the reduction map $\\mathrm{SL}_2(\\Z) \\to \\mathrm{SL}_2(\\Z / N\\Z)$:\n$$\\Gamma(N) = \\set{\\gamma \\in \\mathrm{SL}_2(\\Z) : \\gamma \\equiv I \\pmod N}.$$\nA *congruence subgroup* is a [[subgroup|0X1JXT47GA]] $\\Gamma$ of the [[modular group|VEKV1EM77B]] $\\mathrm{SL}_2(\\Z)$ such that $\\Gamma(N) \\subseteq \\Gamma$ for some $N \\ge 1$.\n\n## Remarks\n\nExample: $\\Gamma_0(N) = \\set{\\begin{pmatrix} a \u0026 b \\\\ c \u0026 d \\end{pmatrix} \\in \\mathrm{SL}_2(\\Z) : c \\equiv 0 \\pmod N}$ is a congruence subgroup, since each matrix of $\\Gamma(N)$ has $c \\equiv 0 \\pmod N$.\nAnother example is $\\Gamma_1(N)$, defined by $c \\equiv 0$ and $a \\equiv d \\equiv 1 \\pmod N$.\nSince $\\mathrm{SL}_2(\\Z / N\\Z)$ is finite, $\\Gamma(N)$ has finite [[index|0DTXBQPA2S]], and so does every congruence subgroup.\nNon-example: the subgroup $\\set{\\begin{pmatrix} 1 \u0026 n \\\\ 0 \u0026 1 \\end{pmatrix} : n \\in \\Z}$ has infinite index, so it contains no $\\Gamma(N)$.\nFor a prime $p$, the index of $\\Gamma_0(p)$ in $\\mathrm{SL}_2(\\Z)$ is $p + 1$.\nNot every subgroup of finite index in $\\mathrm{SL}_2(\\Z)$ is a congruence subgroup.\n",
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