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  "body": "\n## Summary\n\nA congruence subgroup of the [[modular group|VEKV1EM77B]] $\\mathrm{SL}_2(\\Z)$ is a subgroup defined by congruence conditions on the matrix entries modulo some integer $N$.\nThe main examples are $\\Gamma(N)$, $\\Gamma_1(N)$ and $\\Gamma_0(N)$.\n[[Modular forms|GWKN5VG78A]] of level $N$ are modular forms for these groups.\nMost modular forms that arise in arithmetic, such as those attached to [[elliptic curves|427N6S6QQW]], have a congruence subgroup as their symmetry group.\n\n## Explanations\n\n### L1 human\n\nTake the integer matrices of determinant $1$ and reduce their entries modulo $N$.\nA congruence subgroup is a subgroup that can be described by conditions on these reductions.\nFor example, $\\Gamma_0(N)$ is the set of matrices whose lower left entry is divisible by $N$.\nSuch a group is smaller than $\\mathrm{SL}_2(\\Z)$, so functions symmetric under it are easier to find.\n",
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