Properties

Modulus 113
Structure \(C_{112}\)
Order 112

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Show commands for: SageMath / Pari/GP

sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(113)
pari: g = idealstar(,113,2)

Character group

sage: G.order()
pari: g.no
Order = 112
sage: H.invariants()
pari: g.cyc
Structure = \(C_{112}\)
sage: H.gens()
pari: g.gen
Generators = $\chi_{113}(3,\cdot)$

First 32 of 112 characters

Each row describes a character. When available, the columns show the orbit label, order of the character, whether the character is primitive, and several values of the character.

orbit label order primitive -1 1 2 3 4 5 6 7 8 9 10 11
\(\chi_{113}(1,\cdot)\) 113.a 1 No \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{113}(2,\cdot)\) 113.h 28 Yes \(1\) \(1\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{13}{14}\right)\)
\(\chi_{113}(3,\cdot)\) 113.j 112 Yes \(-1\) \(1\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{1}{112}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{83}{112}\right)\) \(e\left(\frac{13}{112}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{1}{56}\right)\) \(e\left(\frac{95}{112}\right)\) \(e\left(\frac{37}{56}\right)\)
\(\chi_{113}(4,\cdot)\) 113.f 14 Yes \(1\) \(1\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{6}{7}\right)\)
\(\chi_{113}(5,\cdot)\) 113.j 112 Yes \(-1\) \(1\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{83}{112}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{57}{112}\right)\) \(e\left(\frac{71}{112}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{27}{56}\right)\) \(e\left(\frac{45}{112}\right)\) \(e\left(\frac{47}{56}\right)\)
\(\chi_{113}(6,\cdot)\) 113.j 112 Yes \(-1\) \(1\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{13}{112}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{71}{112}\right)\) \(e\left(\frac{57}{112}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{13}{56}\right)\) \(e\left(\frac{3}{112}\right)\) \(e\left(\frac{33}{56}\right)\)
\(\chi_{113}(7,\cdot)\) 113.f 14 Yes \(1\) \(1\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{2}{7}\right)\)
\(\chi_{113}(8,\cdot)\) 113.h 28 Yes \(1\) \(1\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{11}{14}\right)\)
\(\chi_{113}(9,\cdot)\) 113.i 56 Yes \(1\) \(1\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{1}{56}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{27}{56}\right)\) \(e\left(\frac{13}{56}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{39}{56}\right)\) \(e\left(\frac{9}{28}\right)\)
\(\chi_{113}(10,\cdot)\) 113.j 112 Yes \(-1\) \(1\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{95}{112}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{45}{112}\right)\) \(e\left(\frac{3}{112}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{39}{56}\right)\) \(e\left(\frac{65}{112}\right)\) \(e\left(\frac{43}{56}\right)\)
\(\chi_{113}(11,\cdot)\) 113.i 56 Yes \(1\) \(1\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{37}{56}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{47}{56}\right)\) \(e\left(\frac{33}{56}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{43}{56}\right)\) \(e\left(\frac{25}{28}\right)\)
\(\chi_{113}(12,\cdot)\) 113.j 112 Yes \(-1\) \(1\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{25}{112}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{59}{112}\right)\) \(e\left(\frac{101}{112}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{25}{56}\right)\) \(e\left(\frac{23}{112}\right)\) \(e\left(\frac{29}{56}\right)\)
\(\chi_{113}(13,\cdot)\) 113.i 56 Yes \(1\) \(1\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{11}{56}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{17}{56}\right)\) \(e\left(\frac{31}{56}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{37}{56}\right)\) \(e\left(\frac{15}{28}\right)\)
\(\chi_{113}(14,\cdot)\) 113.h 28 Yes \(1\) \(1\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{3}{14}\right)\)
\(\chi_{113}(15,\cdot)\) 113.c 4 Yes \(1\) \(1\) \(1\) \(-i\) \(1\) \(i\) \(-i\) \(1\) \(1\) \(-1\) \(i\) \(-1\)
\(\chi_{113}(16,\cdot)\) 113.d 7 Yes \(1\) \(1\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{5}{7}\right)\)
\(\chi_{113}(17,\cdot)\) 113.j 112 Yes \(-1\) \(1\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{5}{112}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{79}{112}\right)\) \(e\left(\frac{65}{112}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{5}{56}\right)\) \(e\left(\frac{27}{112}\right)\) \(e\left(\frac{17}{56}\right)\)
\(\chi_{113}(18,\cdot)\) 113.e 8 Yes \(1\) \(1\) \(-1\) \(e\left(\frac{1}{8}\right)\) \(1\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(1\) \(-1\) \(i\) \(e\left(\frac{7}{8}\right)\) \(i\)
\(\chi_{113}(19,\cdot)\) 113.j 112 Yes \(-1\) \(1\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{99}{112}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{41}{112}\right)\) \(e\left(\frac{55}{112}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{43}{56}\right)\) \(e\left(\frac{109}{112}\right)\) \(e\left(\frac{23}{56}\right)\)
\(\chi_{113}(20,\cdot)\) 113.j 112 Yes \(-1\) \(1\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{107}{112}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{33}{112}\right)\) \(e\left(\frac{47}{112}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{51}{56}\right)\) \(e\left(\frac{85}{112}\right)\) \(e\left(\frac{39}{56}\right)\)
\(\chi_{113}(21,\cdot)\) 113.j 112 Yes \(-1\) \(1\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{9}{112}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{75}{112}\right)\) \(e\left(\frac{5}{112}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{9}{56}\right)\) \(e\left(\frac{71}{112}\right)\) \(e\left(\frac{53}{56}\right)\)
\(\chi_{113}(22,\cdot)\) 113.i 56 Yes \(1\) \(1\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{43}{56}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{41}{56}\right)\) \(e\left(\frac{55}{56}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{53}{56}\right)\) \(e\left(\frac{23}{28}\right)\)
\(\chi_{113}(23,\cdot)\) 113.j 112 Yes \(-1\) \(1\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{41}{112}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{43}{112}\right)\) \(e\left(\frac{85}{112}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{41}{56}\right)\) \(e\left(\frac{87}{112}\right)\) \(e\left(\frac{5}{56}\right)\)
\(\chi_{113}(24,\cdot)\) 113.j 112 Yes \(-1\) \(1\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{37}{112}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{47}{112}\right)\) \(e\left(\frac{33}{112}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{37}{56}\right)\) \(e\left(\frac{43}{112}\right)\) \(e\left(\frac{25}{56}\right)\)
\(\chi_{113}(25,\cdot)\) 113.i 56 Yes \(1\) \(1\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{27}{56}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{1}{56}\right)\) \(e\left(\frac{15}{56}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{45}{56}\right)\) \(e\left(\frac{19}{28}\right)\)
\(\chi_{113}(26,\cdot)\) 113.i 56 Yes \(1\) \(1\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{17}{56}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{11}{56}\right)\) \(e\left(\frac{53}{56}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{47}{56}\right)\) \(e\left(\frac{13}{28}\right)\)
\(\chi_{113}(27,\cdot)\) 113.j 112 Yes \(-1\) \(1\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{3}{112}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{25}{112}\right)\) \(e\left(\frac{39}{112}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{3}{56}\right)\) \(e\left(\frac{61}{112}\right)\) \(e\left(\frac{55}{56}\right)\)
\(\chi_{113}(28,\cdot)\) 113.d 7 Yes \(1\) \(1\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{1}{7}\right)\)
\(\chi_{113}(29,\cdot)\) 113.j 112 Yes \(-1\) \(1\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{89}{112}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{107}{112}\right)\) \(e\left(\frac{37}{112}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{33}{56}\right)\) \(e\left(\frac{55}{112}\right)\) \(e\left(\frac{45}{56}\right)\)
\(\chi_{113}(30,\cdot)\) 113.d 7 Yes \(1\) \(1\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{3}{7}\right)\)
\(\chi_{113}(31,\cdot)\) 113.i 56 Yes \(1\) \(1\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{25}{56}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{3}{56}\right)\) \(e\left(\frac{45}{56}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{23}{56}\right)\) \(e\left(\frac{1}{28}\right)\)
\(\chi_{113}(32,\cdot)\) 113.h 28 Yes \(1\) \(1\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{9}{14}\right)\)