Properties

Modulus 137
Structure \(C_{136}\)
Order 136

Learn more about

Show commands for: SageMath / Pari/GP

sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(137)
pari: g = idealstar(,137,2)

Character group

sage: G.order()
pari: g.no
Order = 136
sage: H.invariants()
pari: g.cyc
Structure = \(C_{136}\)
sage: H.gens()
pari: g.gen
Generators = $\chi_{137}(3,\cdot)$

First 32 of 136 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
\(\chi_{137}(1,\cdot)\) 137.a 1 No \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{137}(2,\cdot)\) 137.g 68 Yes \(1\) \(1\) \(e\left(\frac{25}{34}\right)\) \(e\left(\frac{5}{68}\right)\) \(e\left(\frac{8}{17}\right)\) \(e\left(\frac{35}{68}\right)\) \(e\left(\frac{55}{68}\right)\) \(e\left(\frac{3}{34}\right)\) \(e\left(\frac{7}{34}\right)\) \(e\left(\frac{5}{34}\right)\) \(i\) \(e\left(\frac{33}{34}\right)\)
\(\chi_{137}(3,\cdot)\) 137.h 136 Yes \(-1\) \(1\) \(e\left(\frac{5}{68}\right)\) \(e\left(\frac{1}{136}\right)\) \(e\left(\frac{5}{34}\right)\) \(e\left(\frac{75}{136}\right)\) \(e\left(\frac{11}{136}\right)\) \(e\left(\frac{21}{68}\right)\) \(e\left(\frac{15}{68}\right)\) \(e\left(\frac{1}{68}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{61}{68}\right)\)
\(\chi_{137}(4,\cdot)\) 137.f 34 Yes \(1\) \(1\) \(e\left(\frac{8}{17}\right)\) \(e\left(\frac{5}{34}\right)\) \(e\left(\frac{16}{17}\right)\) \(e\left(\frac{1}{34}\right)\) \(e\left(\frac{21}{34}\right)\) \(e\left(\frac{3}{17}\right)\) \(e\left(\frac{7}{17}\right)\) \(e\left(\frac{5}{17}\right)\) \(-1\) \(e\left(\frac{16}{17}\right)\)
\(\chi_{137}(5,\cdot)\) 137.h 136 Yes \(-1\) \(1\) \(e\left(\frac{35}{68}\right)\) \(e\left(\frac{75}{136}\right)\) \(e\left(\frac{1}{34}\right)\) \(e\left(\frac{49}{136}\right)\) \(e\left(\frac{9}{136}\right)\) \(e\left(\frac{11}{68}\right)\) \(e\left(\frac{37}{68}\right)\) \(e\left(\frac{7}{68}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{19}{68}\right)\)
\(\chi_{137}(6,\cdot)\) 137.h 136 Yes \(-1\) \(1\) \(e\left(\frac{55}{68}\right)\) \(e\left(\frac{11}{136}\right)\) \(e\left(\frac{21}{34}\right)\) \(e\left(\frac{9}{136}\right)\) \(e\left(\frac{121}{136}\right)\) \(e\left(\frac{27}{68}\right)\) \(e\left(\frac{29}{68}\right)\) \(e\left(\frac{11}{68}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{59}{68}\right)\)
\(\chi_{137}(7,\cdot)\) 137.g 68 Yes \(1\) \(1\) \(e\left(\frac{3}{34}\right)\) \(e\left(\frac{21}{68}\right)\) \(e\left(\frac{3}{17}\right)\) \(e\left(\frac{11}{68}\right)\) \(e\left(\frac{27}{68}\right)\) \(e\left(\frac{33}{34}\right)\) \(e\left(\frac{9}{34}\right)\) \(e\left(\frac{21}{34}\right)\) \(i\) \(e\left(\frac{23}{34}\right)\)
\(\chi_{137}(8,\cdot)\) 137.g 68 Yes \(1\) \(1\) \(e\left(\frac{7}{34}\right)\) \(e\left(\frac{15}{68}\right)\) \(e\left(\frac{7}{17}\right)\) \(e\left(\frac{37}{68}\right)\) \(e\left(\frac{29}{68}\right)\) \(e\left(\frac{9}{34}\right)\) \(e\left(\frac{21}{34}\right)\) \(e\left(\frac{15}{34}\right)\) \(-i\) \(e\left(\frac{31}{34}\right)\)
\(\chi_{137}(9,\cdot)\) 137.g 68 Yes \(1\) \(1\) \(e\left(\frac{5}{34}\right)\) \(e\left(\frac{1}{68}\right)\) \(e\left(\frac{5}{17}\right)\) \(e\left(\frac{7}{68}\right)\) \(e\left(\frac{11}{68}\right)\) \(e\left(\frac{21}{34}\right)\) \(e\left(\frac{15}{34}\right)\) \(e\left(\frac{1}{34}\right)\) \(i\) \(e\left(\frac{27}{34}\right)\)
\(\chi_{137}(10,\cdot)\) 137.d 8 Yes \(-1\) \(1\) \(i\) \(e\left(\frac{5}{8}\right)\) \(-1\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(i\) \(-i\) \(i\) \(e\left(\frac{1}{8}\right)\) \(i\)
\(\chi_{137}(11,\cdot)\) 137.g 68 Yes \(1\) \(1\) \(e\left(\frac{33}{34}\right)\) \(e\left(\frac{61}{68}\right)\) \(e\left(\frac{16}{17}\right)\) \(e\left(\frac{19}{68}\right)\) \(e\left(\frac{59}{68}\right)\) \(e\left(\frac{23}{34}\right)\) \(e\left(\frac{31}{34}\right)\) \(e\left(\frac{27}{34}\right)\) \(i\) \(e\left(\frac{15}{34}\right)\)
\(\chi_{137}(12,\cdot)\) 137.h 136 Yes \(-1\) \(1\) \(e\left(\frac{37}{68}\right)\) \(e\left(\frac{21}{136}\right)\) \(e\left(\frac{3}{34}\right)\) \(e\left(\frac{79}{136}\right)\) \(e\left(\frac{95}{136}\right)\) \(e\left(\frac{33}{68}\right)\) \(e\left(\frac{43}{68}\right)\) \(e\left(\frac{21}{68}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{57}{68}\right)\)
\(\chi_{137}(13,\cdot)\) 137.h 136 Yes \(-1\) \(1\) \(e\left(\frac{57}{68}\right)\) \(e\left(\frac{25}{136}\right)\) \(e\left(\frac{23}{34}\right)\) \(e\left(\frac{107}{136}\right)\) \(e\left(\frac{3}{136}\right)\) \(e\left(\frac{49}{68}\right)\) \(e\left(\frac{35}{68}\right)\) \(e\left(\frac{25}{68}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{29}{68}\right)\)
\(\chi_{137}(14,\cdot)\) 137.f 34 Yes \(1\) \(1\) \(e\left(\frac{14}{17}\right)\) \(e\left(\frac{13}{34}\right)\) \(e\left(\frac{11}{17}\right)\) \(e\left(\frac{23}{34}\right)\) \(e\left(\frac{7}{34}\right)\) \(e\left(\frac{1}{17}\right)\) \(e\left(\frac{8}{17}\right)\) \(e\left(\frac{13}{17}\right)\) \(-1\) \(e\left(\frac{11}{17}\right)\)
\(\chi_{137}(15,\cdot)\) 137.f 34 Yes \(1\) \(1\) \(e\left(\frac{10}{17}\right)\) \(e\left(\frac{19}{34}\right)\) \(e\left(\frac{3}{17}\right)\) \(e\left(\frac{31}{34}\right)\) \(e\left(\frac{5}{34}\right)\) \(e\left(\frac{8}{17}\right)\) \(e\left(\frac{13}{17}\right)\) \(e\left(\frac{2}{17}\right)\) \(-1\) \(e\left(\frac{3}{17}\right)\)
\(\chi_{137}(16,\cdot)\) 137.e 17 Yes \(1\) \(1\) \(e\left(\frac{16}{17}\right)\) \(e\left(\frac{5}{17}\right)\) \(e\left(\frac{15}{17}\right)\) \(e\left(\frac{1}{17}\right)\) \(e\left(\frac{4}{17}\right)\) \(e\left(\frac{6}{17}\right)\) \(e\left(\frac{14}{17}\right)\) \(e\left(\frac{10}{17}\right)\) \(1\) \(e\left(\frac{15}{17}\right)\)
\(\chi_{137}(17,\cdot)\) 137.g 68 Yes \(1\) \(1\) \(e\left(\frac{27}{34}\right)\) \(e\left(\frac{19}{68}\right)\) \(e\left(\frac{10}{17}\right)\) \(e\left(\frac{65}{68}\right)\) \(e\left(\frac{5}{68}\right)\) \(e\left(\frac{25}{34}\right)\) \(e\left(\frac{13}{34}\right)\) \(e\left(\frac{19}{34}\right)\) \(-i\) \(e\left(\frac{3}{34}\right)\)
\(\chi_{137}(18,\cdot)\) 137.f 34 Yes \(1\) \(1\) \(e\left(\frac{15}{17}\right)\) \(e\left(\frac{3}{34}\right)\) \(e\left(\frac{13}{17}\right)\) \(e\left(\frac{21}{34}\right)\) \(e\left(\frac{33}{34}\right)\) \(e\left(\frac{12}{17}\right)\) \(e\left(\frac{11}{17}\right)\) \(e\left(\frac{3}{17}\right)\) \(-1\) \(e\left(\frac{13}{17}\right)\)
\(\chi_{137}(19,\cdot)\) 137.g 68 Yes \(1\) \(1\) \(e\left(\frac{13}{34}\right)\) \(e\left(\frac{23}{68}\right)\) \(e\left(\frac{13}{17}\right)\) \(e\left(\frac{25}{68}\right)\) \(e\left(\frac{49}{68}\right)\) \(e\left(\frac{7}{34}\right)\) \(e\left(\frac{5}{34}\right)\) \(e\left(\frac{23}{34}\right)\) \(-i\) \(e\left(\frac{9}{34}\right)\)
\(\chi_{137}(20,\cdot)\) 137.h 136 Yes \(-1\) \(1\) \(e\left(\frac{67}{68}\right)\) \(e\left(\frac{95}{136}\right)\) \(e\left(\frac{33}{34}\right)\) \(e\left(\frac{53}{136}\right)\) \(e\left(\frac{93}{136}\right)\) \(e\left(\frac{23}{68}\right)\) \(e\left(\frac{65}{68}\right)\) \(e\left(\frac{27}{68}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{15}{68}\right)\)
\(\chi_{137}(21,\cdot)\) 137.h 136 Yes \(-1\) \(1\) \(e\left(\frac{11}{68}\right)\) \(e\left(\frac{43}{136}\right)\) \(e\left(\frac{11}{34}\right)\) \(e\left(\frac{97}{136}\right)\) \(e\left(\frac{65}{136}\right)\) \(e\left(\frac{19}{68}\right)\) \(e\left(\frac{33}{68}\right)\) \(e\left(\frac{43}{68}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{39}{68}\right)\)
\(\chi_{137}(22,\cdot)\) 137.f 34 Yes \(1\) \(1\) \(e\left(\frac{12}{17}\right)\) \(e\left(\frac{33}{34}\right)\) \(e\left(\frac{7}{17}\right)\) \(e\left(\frac{27}{34}\right)\) \(e\left(\frac{23}{34}\right)\) \(e\left(\frac{13}{17}\right)\) \(e\left(\frac{2}{17}\right)\) \(e\left(\frac{16}{17}\right)\) \(-1\) \(e\left(\frac{7}{17}\right)\)
\(\chi_{137}(23,\cdot)\) 137.h 136 Yes \(-1\) \(1\) \(e\left(\frac{13}{68}\right)\) \(e\left(\frac{125}{136}\right)\) \(e\left(\frac{13}{34}\right)\) \(e\left(\frac{127}{136}\right)\) \(e\left(\frac{15}{136}\right)\) \(e\left(\frac{41}{68}\right)\) \(e\left(\frac{39}{68}\right)\) \(e\left(\frac{57}{68}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{9}{68}\right)\)
\(\chi_{137}(24,\cdot)\) 137.h 136 Yes \(-1\) \(1\) \(e\left(\frac{19}{68}\right)\) \(e\left(\frac{31}{136}\right)\) \(e\left(\frac{19}{34}\right)\) \(e\left(\frac{13}{136}\right)\) \(e\left(\frac{69}{136}\right)\) \(e\left(\frac{39}{68}\right)\) \(e\left(\frac{57}{68}\right)\) \(e\left(\frac{31}{68}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{55}{68}\right)\)
\(\chi_{137}(25,\cdot)\) 137.g 68 Yes \(1\) \(1\) \(e\left(\frac{1}{34}\right)\) \(e\left(\frac{7}{68}\right)\) \(e\left(\frac{1}{17}\right)\) \(e\left(\frac{49}{68}\right)\) \(e\left(\frac{9}{68}\right)\) \(e\left(\frac{11}{34}\right)\) \(e\left(\frac{3}{34}\right)\) \(e\left(\frac{7}{34}\right)\) \(-i\) \(e\left(\frac{19}{34}\right)\)
\(\chi_{137}(26,\cdot)\) 137.h 136 Yes \(-1\) \(1\) \(e\left(\frac{39}{68}\right)\) \(e\left(\frac{35}{136}\right)\) \(e\left(\frac{5}{34}\right)\) \(e\left(\frac{41}{136}\right)\) \(e\left(\frac{113}{136}\right)\) \(e\left(\frac{55}{68}\right)\) \(e\left(\frac{49}{68}\right)\) \(e\left(\frac{35}{68}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{27}{68}\right)\)
\(\chi_{137}(27,\cdot)\) 137.h 136 Yes \(-1\) \(1\) \(e\left(\frac{15}{68}\right)\) \(e\left(\frac{3}{136}\right)\) \(e\left(\frac{15}{34}\right)\) \(e\left(\frac{89}{136}\right)\) \(e\left(\frac{33}{136}\right)\) \(e\left(\frac{63}{68}\right)\) \(e\left(\frac{45}{68}\right)\) \(e\left(\frac{3}{68}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{47}{68}\right)\)
\(\chi_{137}(28,\cdot)\) 137.g 68 Yes \(1\) \(1\) \(e\left(\frac{19}{34}\right)\) \(e\left(\frac{31}{68}\right)\) \(e\left(\frac{2}{17}\right)\) \(e\left(\frac{13}{68}\right)\) \(e\left(\frac{1}{68}\right)\) \(e\left(\frac{5}{34}\right)\) \(e\left(\frac{23}{34}\right)\) \(e\left(\frac{31}{34}\right)\) \(-i\) \(e\left(\frac{21}{34}\right)\)
\(\chi_{137}(29,\cdot)\) 137.h 136 Yes \(-1\) \(1\) \(e\left(\frac{47}{68}\right)\) \(e\left(\frac{91}{136}\right)\) \(e\left(\frac{13}{34}\right)\) \(e\left(\frac{25}{136}\right)\) \(e\left(\frac{49}{136}\right)\) \(e\left(\frac{7}{68}\right)\) \(e\left(\frac{5}{68}\right)\) \(e\left(\frac{23}{68}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{43}{68}\right)\)
\(\chi_{137}(30,\cdot)\) 137.g 68 Yes \(1\) \(1\) \(e\left(\frac{11}{34}\right)\) \(e\left(\frac{43}{68}\right)\) \(e\left(\frac{11}{17}\right)\) \(e\left(\frac{29}{68}\right)\) \(e\left(\frac{65}{68}\right)\) \(e\left(\frac{19}{34}\right)\) \(e\left(\frac{33}{34}\right)\) \(e\left(\frac{9}{34}\right)\) \(-i\) \(e\left(\frac{5}{34}\right)\)
\(\chi_{137}(31,\cdot)\) 137.h 136 Yes \(-1\) \(1\) \(e\left(\frac{25}{68}\right)\) \(e\left(\frac{73}{136}\right)\) \(e\left(\frac{25}{34}\right)\) \(e\left(\frac{35}{136}\right)\) \(e\left(\frac{123}{136}\right)\) \(e\left(\frac{37}{68}\right)\) \(e\left(\frac{7}{68}\right)\) \(e\left(\frac{5}{68}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{33}{68}\right)\)
\(\chi_{137}(32,\cdot)\) 137.g 68 Yes \(1\) \(1\) \(e\left(\frac{23}{34}\right)\) \(e\left(\frac{25}{68}\right)\) \(e\left(\frac{6}{17}\right)\) \(e\left(\frac{39}{68}\right)\) \(e\left(\frac{3}{68}\right)\) \(e\left(\frac{15}{34}\right)\) \(e\left(\frac{1}{34}\right)\) \(e\left(\frac{25}{34}\right)\) \(i\) \(e\left(\frac{29}{34}\right)\)