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The ModularGroup Package

Finite-index subgroups of \({\rm (P)SL}_2(ℤ)\)

Version 2.0.2

4 May 2026

Sebastian Engelhardt
Email: seen00001@stud.uni-saarland.de
Homepage: https://www.uni-saarland.de/lehrstuhl/weitze-schmithuesen.html

Luca L. Junk
Email: junk@math.uni-sb.de
Homepage: https://www.uni-saarland.de/lehrstuhl/weber-moritz/team/luca-junk.html

Hannah Wagmann
Email: wagmann@math.uni-sb.de
Homepage: https://www.uni-saarland.de/lehrstuhl/weitze-schmithuesen/team/hannah-wagmann.html

Gabriela Weitze-Schmithüsen
Email: weitze@math.uni-sb.de
Homepage: https://www.uni-saarland.de/lehrstuhl/weitze-schmithuesen/team/gabriela-weitze-schmithuesen.html

Copyright

© 2026 by Sebastian Engelhardt, Luca L. Junk, Hannah Wagmann and Gabriela Weitze-Schmithüsen

Acknowledgements

Supported by Project I.8 of SFB-TRR 195 'Symbolic Tools in Mathematics and their Application' of the German Research Foundation (DFG).

Contents

1 Introduction
 1.1 General aims of the ModularGroup package
 1.2 Technicalities
2 Subgroups of \({\rm SL}_2(ℤ)\)
 2.1 Construction of modular subgroups

  2.1-1 ModularSubgroup
  2.1-2 ModularSubgroupST
  2.1-3 ModularSubgroupRT
  2.1-4 ModularSubgroupSJ
  2.1-5 ModularSubgroup
 2.2 Getters for the coset action

  2.2-1 SAction
  2.2-2 TAction
  2.2-3 RAction
  2.2-4 JAction
  2.2-5 CosetActionOf
 2.3 Computing with modular subgroups

  2.3-1 Index
  2.3-2 GeneralizedLevel
  2.3-3 CosetRepresentatives
  2.3-4 WordGeneratorsOfGroup
  2.3-5 GeneratorsOfGroup
  2.3-6 IsCongruence
  2.3-7 Cusps
  2.3-8 CuspWidth
  2.3-9 CuspsEquivalent
  2.3-10 CosetRepresentativeOfCusp
  2.3-11 IndexModN
  2.3-12 Deficiency
  2.3-13 Deficiency
  2.3-14 Projectivization
  2.3-15 ConjugateGroup
  2.3-16 NormalCore
  2.3-17 QuotientByNormalCore
  2.3-18 AssociatedCharacterTable
  2.3-19 IsElementOf
  2.3-20 IsWordElementOf
  2.3-21 IsWordElementOf
  2.3-22 Genus
 2.4 Miscellaneous

  2.4-1 DefinesCosetActionST
  2.4-2 DefinesCosetActionRT
  2.4-3 DefinesCosetActionSJ
  2.4-4 CosetActionFromGenerators
  2.4-5 STDecomposition
  2.4-6 RTDecomposition
  2.4-7 SJDecomposition
  2.4-8 STDecompositionAsList
3 Subgroups of \({\rm PSL}_2(ℤ)\)
 3.1 Construction of projective modular subgroups

  3.1-1 ProjectiveModularSubgroup
 3.2 Getters for the coset action

  3.2-1 SAction
  3.2-2 TAction
 3.3 Computing with projective modular subgroups

  3.3-1 Index
  3.3-2 GeneralizedLevel
  3.3-3 CosetRepresentatives
  3.3-4 WordGeneratorsOfGroup
  3.3-5 IsCongruence
  3.3-6 Cusps
  3.3-7 CuspWidth
  3.3-8 CuspsEquivalent
  3.3-9 CosetRepresentativeOfCusp
  3.3-10 LiftToSL2ZEven
  3.3-11 LiftToSL2ZOdd
  3.3-12 IndexModN
  3.3-13 Deficiency
  3.3-14 Deficiency
  3.3-15 ConjugateGroup
  3.3-16 NormalCore
  3.3-17 QuotientByNormalCore
  3.3-18 AssociatedCharacterTable
  3.3-19 IsElementOf
  3.3-20 IsWordElementOf
  3.3-21 IsWordElementOf
  3.3-22 Genus
References
Index

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