# Properties

 Modulus $169$ Structure $$C_{156}$$ Order $156$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(169)

pari: g = idealstar(,169,2)

## Character group

 sage: G.order()  pari: g.no Order = 156 sage: H.invariants()  pari: g.cyc Structure = $$C_{156}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{169}(2,\cdot)$

## First 32 of 156 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 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$
$$\chi_{169}(1,\cdot)$$ 169.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{169}(2,\cdot)$$ 169.l 156 yes $$-1$$ $$1$$ $$e\left(\frac{1}{156}\right)$$ $$e\left(\frac{31}{39}\right)$$ $$e\left(\frac{1}{78}\right)$$ $$e\left(\frac{3}{52}\right)$$ $$e\left(\frac{125}{156}\right)$$ $$e\left(\frac{107}{156}\right)$$ $$e\left(\frac{1}{52}\right)$$ $$e\left(\frac{23}{39}\right)$$ $$e\left(\frac{5}{78}\right)$$ $$e\left(\frac{103}{156}\right)$$
$$\chi_{169}(3,\cdot)$$ 169.i 39 yes $$1$$ $$1$$ $$e\left(\frac{31}{39}\right)$$ $$e\left(\frac{22}{39}\right)$$ $$e\left(\frac{23}{39}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{14}{39}\right)$$ $$e\left(\frac{2}{39}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{5}{39}\right)$$ $$e\left(\frac{37}{39}\right)$$ $$e\left(\frac{34}{39}\right)$$
$$\chi_{169}(4,\cdot)$$ 169.k 78 yes $$1$$ $$1$$ $$e\left(\frac{1}{78}\right)$$ $$e\left(\frac{23}{39}\right)$$ $$e\left(\frac{1}{39}\right)$$ $$e\left(\frac{3}{26}\right)$$ $$e\left(\frac{47}{78}\right)$$ $$e\left(\frac{29}{78}\right)$$ $$e\left(\frac{1}{26}\right)$$ $$e\left(\frac{7}{39}\right)$$ $$e\left(\frac{5}{39}\right)$$ $$e\left(\frac{25}{78}\right)$$
$$\chi_{169}(5,\cdot)$$ 169.j 52 yes $$-1$$ $$1$$ $$e\left(\frac{3}{52}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{3}{26}\right)$$ $$e\left(\frac{27}{52}\right)$$ $$e\left(\frac{11}{52}\right)$$ $$e\left(\frac{9}{52}\right)$$ $$e\left(\frac{9}{52}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{15}{26}\right)$$ $$e\left(\frac{49}{52}\right)$$
$$\chi_{169}(6,\cdot)$$ 169.l 156 yes $$-1$$ $$1$$ $$e\left(\frac{125}{156}\right)$$ $$e\left(\frac{14}{39}\right)$$ $$e\left(\frac{47}{78}\right)$$ $$e\left(\frac{11}{52}\right)$$ $$e\left(\frac{25}{156}\right)$$ $$e\left(\frac{115}{156}\right)$$ $$e\left(\frac{21}{52}\right)$$ $$e\left(\frac{28}{39}\right)$$ $$e\left(\frac{1}{78}\right)$$ $$e\left(\frac{83}{156}\right)$$
$$\chi_{169}(7,\cdot)$$ 169.l 156 yes $$-1$$ $$1$$ $$e\left(\frac{107}{156}\right)$$ $$e\left(\frac{2}{39}\right)$$ $$e\left(\frac{29}{78}\right)$$ $$e\left(\frac{9}{52}\right)$$ $$e\left(\frac{115}{156}\right)$$ $$e\left(\frac{61}{156}\right)$$ $$e\left(\frac{3}{52}\right)$$ $$e\left(\frac{4}{39}\right)$$ $$e\left(\frac{67}{78}\right)$$ $$e\left(\frac{101}{156}\right)$$
$$\chi_{169}(8,\cdot)$$ 169.j 52 yes $$-1$$ $$1$$ $$e\left(\frac{1}{52}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{1}{26}\right)$$ $$e\left(\frac{9}{52}\right)$$ $$e\left(\frac{21}{52}\right)$$ $$e\left(\frac{3}{52}\right)$$ $$e\left(\frac{3}{52}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{5}{26}\right)$$ $$e\left(\frac{51}{52}\right)$$
$$\chi_{169}(9,\cdot)$$ 169.i 39 yes $$1$$ $$1$$ $$e\left(\frac{23}{39}\right)$$ $$e\left(\frac{5}{39}\right)$$ $$e\left(\frac{7}{39}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{28}{39}\right)$$ $$e\left(\frac{4}{39}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{10}{39}\right)$$ $$e\left(\frac{35}{39}\right)$$ $$e\left(\frac{29}{39}\right)$$
$$\chi_{169}(10,\cdot)$$ 169.k 78 yes $$1$$ $$1$$ $$e\left(\frac{5}{78}\right)$$ $$e\left(\frac{37}{39}\right)$$ $$e\left(\frac{5}{39}\right)$$ $$e\left(\frac{15}{26}\right)$$ $$e\left(\frac{1}{78}\right)$$ $$e\left(\frac{67}{78}\right)$$ $$e\left(\frac{5}{26}\right)$$ $$e\left(\frac{35}{39}\right)$$ $$e\left(\frac{25}{39}\right)$$ $$e\left(\frac{47}{78}\right)$$
$$\chi_{169}(11,\cdot)$$ 169.l 156 yes $$-1$$ $$1$$ $$e\left(\frac{103}{156}\right)$$ $$e\left(\frac{34}{39}\right)$$ $$e\left(\frac{25}{78}\right)$$ $$e\left(\frac{49}{52}\right)$$ $$e\left(\frac{83}{156}\right)$$ $$e\left(\frac{101}{156}\right)$$ $$e\left(\frac{51}{52}\right)$$ $$e\left(\frac{29}{39}\right)$$ $$e\left(\frac{47}{78}\right)$$ $$e\left(\frac{1}{156}\right)$$
$$\chi_{169}(12,\cdot)$$ 169.h 26 yes $$1$$ $$1$$ $$e\left(\frac{21}{26}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{8}{13}\right)$$ $$e\left(\frac{7}{26}\right)$$ $$e\left(\frac{25}{26}\right)$$ $$e\left(\frac{11}{26}\right)$$ $$e\left(\frac{11}{26}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{5}{26}\right)$$
$$\chi_{169}(14,\cdot)$$ 169.g 13 yes $$1$$ $$1$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{7}{13}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{4}{13}\right)$$
$$\chi_{169}(15,\cdot)$$ 169.l 156 yes $$-1$$ $$1$$ $$e\left(\frac{133}{156}\right)$$ $$e\left(\frac{28}{39}\right)$$ $$e\left(\frac{55}{78}\right)$$ $$e\left(\frac{35}{52}\right)$$ $$e\left(\frac{89}{156}\right)$$ $$e\left(\frac{35}{156}\right)$$ $$e\left(\frac{29}{52}\right)$$ $$e\left(\frac{17}{39}\right)$$ $$e\left(\frac{41}{78}\right)$$ $$e\left(\frac{127}{156}\right)$$
$$\chi_{169}(16,\cdot)$$ 169.i 39 yes $$1$$ $$1$$ $$e\left(\frac{1}{39}\right)$$ $$e\left(\frac{7}{39}\right)$$ $$e\left(\frac{2}{39}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{8}{39}\right)$$ $$e\left(\frac{29}{39}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{14}{39}\right)$$ $$e\left(\frac{10}{39}\right)$$ $$e\left(\frac{25}{39}\right)$$
$$\chi_{169}(17,\cdot)$$ 169.k 78 yes $$1$$ $$1$$ $$e\left(\frac{73}{78}\right)$$ $$e\left(\frac{2}{39}\right)$$ $$e\left(\frac{34}{39}\right)$$ $$e\left(\frac{11}{26}\right)$$ $$e\left(\frac{77}{78}\right)$$ $$e\left(\frac{11}{78}\right)$$ $$e\left(\frac{21}{26}\right)$$ $$e\left(\frac{4}{39}\right)$$ $$e\left(\frac{14}{39}\right)$$ $$e\left(\frac{31}{78}\right)$$
$$\chi_{169}(18,\cdot)$$ 169.j 52 yes $$-1$$ $$1$$ $$e\left(\frac{31}{52}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{5}{26}\right)$$ $$e\left(\frac{19}{52}\right)$$ $$e\left(\frac{27}{52}\right)$$ $$e\left(\frac{41}{52}\right)$$ $$e\left(\frac{41}{52}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{25}{26}\right)$$ $$e\left(\frac{21}{52}\right)$$
$$\chi_{169}(19,\cdot)$$ 169.f 12 no $$-1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-i$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$i$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$
$$\chi_{169}(20,\cdot)$$ 169.l 156 yes $$-1$$ $$1$$ $$e\left(\frac{11}{156}\right)$$ $$e\left(\frac{29}{39}\right)$$ $$e\left(\frac{11}{78}\right)$$ $$e\left(\frac{33}{52}\right)$$ $$e\left(\frac{127}{156}\right)$$ $$e\left(\frac{85}{156}\right)$$ $$e\left(\frac{11}{52}\right)$$ $$e\left(\frac{19}{39}\right)$$ $$e\left(\frac{55}{78}\right)$$ $$e\left(\frac{41}{156}\right)$$
$$\chi_{169}(21,\cdot)$$ 169.j 52 yes $$-1$$ $$1$$ $$e\left(\frac{25}{52}\right)$$ $$e\left(\frac{8}{13}\right)$$ $$e\left(\frac{25}{26}\right)$$ $$e\left(\frac{17}{52}\right)$$ $$e\left(\frac{5}{52}\right)$$ $$e\left(\frac{23}{52}\right)$$ $$e\left(\frac{23}{52}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{21}{26}\right)$$ $$e\left(\frac{27}{52}\right)$$
$$\chi_{169}(22,\cdot)$$ 169.c 3 no $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{169}(23,\cdot)$$ 169.e 6 no $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{169}(24,\cdot)$$ 169.l 156 yes $$-1$$ $$1$$ $$e\left(\frac{127}{156}\right)$$ $$e\left(\frac{37}{39}\right)$$ $$e\left(\frac{49}{78}\right)$$ $$e\left(\frac{17}{52}\right)$$ $$e\left(\frac{119}{156}\right)$$ $$e\left(\frac{17}{156}\right)$$ $$e\left(\frac{23}{52}\right)$$ $$e\left(\frac{35}{39}\right)$$ $$e\left(\frac{11}{78}\right)$$ $$e\left(\frac{133}{156}\right)$$
$$\chi_{169}(25,\cdot)$$ 169.h 26 yes $$1$$ $$1$$ $$e\left(\frac{3}{26}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{1}{26}\right)$$ $$e\left(\frac{11}{26}\right)$$ $$e\left(\frac{9}{26}\right)$$ $$e\left(\frac{9}{26}\right)$$ $$e\left(\frac{8}{13}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{23}{26}\right)$$
$$\chi_{169}(27,\cdot)$$ 169.g 13 yes $$1$$ $$1$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{8}{13}\right)$$
$$\chi_{169}(28,\cdot)$$ 169.l 156 yes $$-1$$ $$1$$ $$e\left(\frac{109}{156}\right)$$ $$e\left(\frac{25}{39}\right)$$ $$e\left(\frac{31}{78}\right)$$ $$e\left(\frac{15}{52}\right)$$ $$e\left(\frac{53}{156}\right)$$ $$e\left(\frac{119}{156}\right)$$ $$e\left(\frac{5}{52}\right)$$ $$e\left(\frac{11}{39}\right)$$ $$e\left(\frac{77}{78}\right)$$ $$e\left(\frac{151}{156}\right)$$
$$\chi_{169}(29,\cdot)$$ 169.i 39 yes $$1$$ $$1$$ $$e\left(\frac{10}{39}\right)$$ $$e\left(\frac{31}{39}\right)$$ $$e\left(\frac{20}{39}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{2}{39}\right)$$ $$e\left(\frac{17}{39}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{23}{39}\right)$$ $$e\left(\frac{22}{39}\right)$$ $$e\left(\frac{16}{39}\right)$$
$$\chi_{169}(30,\cdot)$$ 169.k 78 yes $$1$$ $$1$$ $$e\left(\frac{67}{78}\right)$$ $$e\left(\frac{20}{39}\right)$$ $$e\left(\frac{28}{39}\right)$$ $$e\left(\frac{19}{26}\right)$$ $$e\left(\frac{29}{78}\right)$$ $$e\left(\frac{71}{78}\right)$$ $$e\left(\frac{15}{26}\right)$$ $$e\left(\frac{1}{39}\right)$$ $$e\left(\frac{23}{39}\right)$$ $$e\left(\frac{37}{78}\right)$$
$$\chi_{169}(31,\cdot)$$ 169.j 52 yes $$-1$$ $$1$$ $$e\left(\frac{7}{52}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{7}{26}\right)$$ $$e\left(\frac{11}{52}\right)$$ $$e\left(\frac{43}{52}\right)$$ $$e\left(\frac{21}{52}\right)$$ $$e\left(\frac{21}{52}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{9}{26}\right)$$ $$e\left(\frac{45}{52}\right)$$
$$\chi_{169}(32,\cdot)$$ 169.l 156 yes $$-1$$ $$1$$ $$e\left(\frac{5}{156}\right)$$ $$e\left(\frac{38}{39}\right)$$ $$e\left(\frac{5}{78}\right)$$ $$e\left(\frac{15}{52}\right)$$ $$e\left(\frac{1}{156}\right)$$ $$e\left(\frac{67}{156}\right)$$ $$e\left(\frac{5}{52}\right)$$ $$e\left(\frac{37}{39}\right)$$ $$e\left(\frac{25}{78}\right)$$ $$e\left(\frac{47}{156}\right)$$
$$\chi_{169}(33,\cdot)$$ 169.l 156 yes $$-1$$ $$1$$ $$e\left(\frac{71}{156}\right)$$ $$e\left(\frac{17}{39}\right)$$ $$e\left(\frac{71}{78}\right)$$ $$e\left(\frac{5}{52}\right)$$ $$e\left(\frac{139}{156}\right)$$ $$e\left(\frac{109}{156}\right)$$ $$e\left(\frac{19}{52}\right)$$ $$e\left(\frac{34}{39}\right)$$ $$e\left(\frac{43}{78}\right)$$ $$e\left(\frac{137}{156}\right)$$
$$\chi_{169}(34,\cdot)$$ 169.j 52 yes $$-1$$ $$1$$ $$e\left(\frac{49}{52}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{23}{26}\right)$$ $$e\left(\frac{25}{52}\right)$$ $$e\left(\frac{41}{52}\right)$$ $$e\left(\frac{43}{52}\right)$$ $$e\left(\frac{43}{52}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{11}{26}\right)$$ $$e\left(\frac{3}{52}\right)$$