# Properties

 Modulus $137$ Structure $$C_{136}$$ Order $136$

Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(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.

Character Orbit 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 $$1$$ $$1$$ $$e\left(\frac{23}{34}\right)$$ $$e\left(\frac{25}{68}\right)$$ $$e\left(\frac{6}{17}\right)$$ $$e\left(\frac{39}{68}\right)$$ $$e\left(\frac{3}{68}\right)$$ $$e\left(\frac{15}{34}\right)$$ $$e\left(\frac{1}{34}\right)$$ $$e\left(\frac{25}{34}\right)$$ $$i$$ $$e\left(\frac{29}{34}\right)$$