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I have found in a paper the group $\Gamma_0(18)$ can be generated by the following list of matrices:

  • $\displaystyle \left( \begin{array}{rr} 7 & -1 \\ 36 & 5 \end{array} \right) $

  • $\displaystyle \left( \begin{array}{rr} 13 & -8 \\ 18 & -11 \end{array} \right) $

  • $\displaystyle \left( \begin{array}{rr} 71 & -15 \\ 90 & -9 \end{array} \right) $

  • $\displaystyle \left( \begin{array}{rr} 55 & -13 \\ 72 & -17 \end{array} \right) $
  • $\displaystyle \left( \begin{array}{rr} 7 & -2 \\ 18 & -5 \end{array} \right) $
  • $\displaystyle \left( \begin{array}{rr} 31 & -25 \\ 36 & -29 \end{array} \right) $
  • $\displaystyle \left( \begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array} \right) $
  • $\displaystyle \left( \begin{array}{rr} -1 & 0 \\ 0 & -1 \end{array} \right) $

E.g. $31 \times (-29) - 36 \times (-25) = -899+900=1 $ In fact, $36, 18, 19 \equiv 0 \pmod {18}$.

These generators look arbitrary. Is there a list of generators of $\Gamma_0(N)$ for $N < 100$ ?

Is there a computer program or algorithm for finding generators of congruence groups, example using a linear algebra package such as sage ?

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    $\begingroup$ By using the known presentation of ${\rm SL}(2,{\mathbb Z})$ and expressing the congruence subgroups as kernels of homomorphisms onto a finite group, you could use standard algorithms (available in GAP or Magma) for computing generators of these kernels as subgroups of a group defined by a finite presentation. I don't know whether that's the easiest way to do it! $\endgroup$
    – Derek Holt
    Mar 10, 2020 at 13:34
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    $\begingroup$ Googling "sage congruence subgroup", the first hit is doc.sagemath.org/html/en/reference/arithgroup/sage/modular/… which tells you how to list generators. The online Sage interface sagecell.sagemath.org lists the generators of $\Gamma_0(N)$ quickly for $N$ up to at least 5000. $\endgroup$
    – Bort
    Mar 10, 2020 at 13:34
  • $\begingroup$ See also this Math.SE question: math.stackexchange.com/questions/1758302/… $\endgroup$
    – David Loeffler
    Mar 11, 2020 at 7:27
  • $\begingroup$ There are also theoretical results for every $N$ (and even presentations of $\Gamma_0(N)$): see this article by Lascurain Orive ams.org/journals/ecgd/2002-06-03/S1088-4173-02-00073-5/… $\endgroup$
    – François Brunault
    Mar 11, 2020 at 9:10
  • $\begingroup$ There are even geometric algorithms for arbitrary finite-index subgroups using Farey symbols. See arxiv.org/abs/1809.04030v2 just after Théorème 1.10. $\endgroup$
    – François Brunault
    Mar 11, 2020 at 9:26

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sage should have what you need.

Check under the documentation for the modular group. Specifically under generators you can find a couple of working examples for finding the generating set for $\Gamma_0(3)$.

I don't think GAP can do this built-in, as an aside.

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    $\begingroup$ The Congruence package in Gap can do this, even for arbitrary finite index subgroups it seems (the requirement is to have a membership test). See gap-system.org/Packages/congruence.html $\endgroup$
    – François Brunault
    Mar 11, 2020 at 9:22
  • $\begingroup$ I see. I couldn't find that one, but that seems to do the trick too. Thanks. $\endgroup$
    – Carl-Fredrik Nyberg Brodda
    Mar 11, 2020 at 12:06

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