\contentsline {chapter}{\numberline {1}\textcolor {Chapter }{Introduction}}{4}{chapter.1}%
\contentsline {section}{\numberline {1.1}\textcolor {Chapter }{General aims of \textsf {Congruence} package}}{4}{section.1.1}%
\contentsline {section}{\numberline {1.2}\textcolor {Chapter }{Installation and system requirements}}{4}{section.1.2}%
\contentsline {chapter}{\numberline {2}\textcolor {Chapter }{Construction of congruence subgroups}}{5}{chapter.2}%
\contentsline {section}{\numberline {2.1}\textcolor {Chapter }{Construction of congruence subgroups}}{5}{section.2.1}%
\contentsline {subsection}{\numberline {2.1.1}\textcolor {Chapter }{PrincipalCongruenceSubgroup}}{5}{subsection.2.1.1}%
\contentsline {subsection}{\numberline {2.1.2}\textcolor {Chapter }{CongruenceSubgroupGamma0}}{6}{subsection.2.1.2}%
\contentsline {subsection}{\numberline {2.1.3}\textcolor {Chapter }{CongruenceSubgroupGammaUpper0}}{6}{subsection.2.1.3}%
\contentsline {subsection}{\numberline {2.1.4}\textcolor {Chapter }{CongruenceSubgroupGamma1}}{7}{subsection.2.1.4}%
\contentsline {subsection}{\numberline {2.1.5}\textcolor {Chapter }{CongruenceSubgroupGammaUpper1}}{7}{subsection.2.1.5}%
\contentsline {subsection}{\numberline {2.1.6}\textcolor {Chapter }{IntersectionOfCongruenceSubgroups}}{7}{subsection.2.1.6}%
\contentsline {section}{\numberline {2.2}\textcolor {Chapter }{Properties of congruence subgroups}}{8}{section.2.2}%
\contentsline {subsection}{\numberline {2.2.1}\textcolor {Chapter }{IsPrincipalCongruenceSubgroup}}{8}{subsection.2.2.1}%
\contentsline {subsection}{\numberline {2.2.2}\textcolor {Chapter }{IsCongruenceSubgroupGamma0}}{8}{subsection.2.2.2}%
\contentsline {subsection}{\numberline {2.2.3}\textcolor {Chapter }{IsCongruenceSubgroupGammaUpper0}}{8}{subsection.2.2.3}%
\contentsline {subsection}{\numberline {2.2.4}\textcolor {Chapter }{IsCongruenceSubgroupGamma1}}{8}{subsection.2.2.4}%
\contentsline {subsection}{\numberline {2.2.5}\textcolor {Chapter }{IsCongruenceSubgroupGammaUpper1}}{9}{subsection.2.2.5}%
\contentsline {subsection}{\numberline {2.2.6}\textcolor {Chapter }{IsIntersectionOfCongruenceSubgroups}}{9}{subsection.2.2.6}%
\contentsline {section}{\numberline {2.3}\textcolor {Chapter }{Attributes of congruence subgroups}}{9}{section.2.3}%
\contentsline {subsection}{\numberline {2.3.1}\textcolor {Chapter }{LevelOfCongruenceSubgroup}}{9}{subsection.2.3.1}%
\contentsline {subsection}{\numberline {2.3.2}\textcolor {Chapter }{IndexInSL2Z}}{10}{subsection.2.3.2}%
\contentsline {subsection}{\numberline {2.3.3}\textcolor {Chapter }{DefiningCongruenceSubgroups}}{10}{subsection.2.3.3}%
\contentsline {section}{\numberline {2.4}\textcolor {Chapter }{Operations for congruence subgroups}}{10}{section.2.4}%
\contentsline {subsection}{\numberline {2.4.1}\textcolor {Chapter }{Random (one and two argument versions)}}{10}{subsection.2.4.1}%
\contentsline {subsection}{\numberline {2.4.2}\textcolor {Chapter }{\texttt {\symbol {92}}in}}{11}{subsection.2.4.2}%
\contentsline {subsection}{\numberline {2.4.3}\textcolor {Chapter }{CanEasilyCompareCongruenceSubgroups}}{11}{subsection.2.4.3}%
\contentsline {subsection}{\numberline {2.4.4}\textcolor {Chapter }{IsSubset}}{11}{subsection.2.4.4}%
\contentsline {subsection}{\numberline {2.4.5}\textcolor {Chapter }{Index}}{12}{subsection.2.4.5}%
\contentsline {chapter}{\numberline {3}\textcolor {Chapter }{Farey symbols and their properties}}{13}{chapter.3}%
\contentsline {section}{\numberline {3.1}\textcolor {Chapter }{Construction of Farey symbols}}{13}{section.3.1}%
\contentsline {subsection}{\numberline {3.1.1}\textcolor {Chapter }{FareySymbolByData}}{13}{subsection.3.1.1}%
\contentsline {subsection}{\numberline {3.1.2}\textcolor {Chapter }{IsValidFareySymbol}}{14}{subsection.3.1.2}%
\contentsline {section}{\numberline {3.2}\textcolor {Chapter }{Properties of Farey symbols}}{14}{section.3.2}%
\contentsline {subsection}{\numberline {3.2.1}\textcolor {Chapter }{GeneralizedFareySequence}}{14}{subsection.3.2.1}%
\contentsline {subsection}{\numberline {3.2.2}\textcolor {Chapter }{NumeratorOfGFSElement}}{14}{subsection.3.2.2}%
\contentsline {subsection}{\numberline {3.2.3}\textcolor {Chapter }{DenominatorOfGFSElement}}{14}{subsection.3.2.3}%
\contentsline {subsection}{\numberline {3.2.4}\textcolor {Chapter }{LabelsOfFareySymbol}}{15}{subsection.3.2.4}%
\contentsline {chapter}{\numberline {4}\textcolor {Chapter }{Farey symbols for congruence subgroups}}{16}{chapter.4}%
\contentsline {section}{\numberline {4.1}\textcolor {Chapter }{Computation of the Farey symbol for a finite index subgroup}}{16}{section.4.1}%
\contentsline {subsection}{\numberline {4.1.1}\textcolor {Chapter }{FareySymbol}}{16}{subsection.4.1.1}%
\contentsline {section}{\numberline {4.2}\textcolor {Chapter }{Computation of generators of a finite index subgroup from its Farey symbol}}{17}{section.4.2}%
\contentsline {subsection}{\numberline {4.2.1}\textcolor {Chapter }{MatrixByEvenInterval}}{17}{subsection.4.2.1}%
\contentsline {subsection}{\numberline {4.2.2}\textcolor {Chapter }{MatrixByOddInterval}}{17}{subsection.4.2.2}%
\contentsline {subsection}{\numberline {4.2.3}\textcolor {Chapter }{MatrixByFreePairOfIntervals}}{18}{subsection.4.2.3}%
\contentsline {subsection}{\numberline {4.2.4}\textcolor {Chapter }{GeneratorsByFareySymbol}}{18}{subsection.4.2.4}%
\contentsline {subsection}{\numberline {4.2.5}\textcolor {Chapter }{GeneratorsOfGroup}}{18}{subsection.4.2.5}%
\contentsline {section}{\numberline {4.3}\textcolor {Chapter }{Other properties derived from Farey symbols}}{19}{section.4.3}%
\contentsline {subsection}{\numberline {4.3.1}\textcolor {Chapter }{IndexInPSL2ZByFareySymbol}}{19}{subsection.4.3.1}%
\contentsline {chapter}{\numberline {5}\textcolor {Chapter }{Service functions of the \textsf {Congruence} package}}{20}{chapter.5}%
\contentsline {section}{\numberline {5.1}\textcolor {Chapter }{Additional information displayed by \textsf {Congruence} algorithms}}{20}{section.5.1}%
\contentsline {subsection}{\numberline {5.1.1}\textcolor {Chapter }{InfoCongruence}}{20}{subsection.5.1.1}%
\contentsline {chapter}{References}{21}{chapter*.5}%
\contentsline {chapter}{Index}{22}{section*.6}%
