# Properties

 Modulus 163 Structure $$C_{162}$$ Order 162

Show commands for: SageMath / Pari/GP

sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed

sage: H = DirichletGroup_conrey(163)

pari: g = idealstar(,163,2)

## Character group

 sage: G.order()  pari: g.no Order = 162 sage: H.invariants()  pari: g.cyc Structure = $$C_{162}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{163}(2,\cdot)$

## First 32 of 162 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
$$\chi_{163}(1,\cdot)$$ 163.a 1 No $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{163}(2,\cdot)$$ 163.j 162 Yes $$-1$$ $$1$$ $$e\left(\frac{1}{162}\right)$$ $$e\left(\frac{101}{162}\right)$$ $$e\left(\frac{1}{81}\right)$$ $$e\left(\frac{5}{54}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{73}{162}\right)$$ $$e\left(\frac{1}{54}\right)$$ $$e\left(\frac{20}{81}\right)$$ $$e\left(\frac{8}{81}\right)$$ $$e\left(\frac{47}{162}\right)$$
$$\chi_{163}(3,\cdot)$$ 163.j 162 Yes $$-1$$ $$1$$ $$e\left(\frac{101}{162}\right)$$ $$e\left(\frac{157}{162}\right)$$ $$e\left(\frac{20}{81}\right)$$ $$e\left(\frac{19}{54}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{83}{162}\right)$$ $$e\left(\frac{47}{54}\right)$$ $$e\left(\frac{76}{81}\right)$$ $$e\left(\frac{79}{81}\right)$$ $$e\left(\frac{49}{162}\right)$$
$$\chi_{163}(4,\cdot)$$ 163.i 81 Yes $$1$$ $$1$$ $$e\left(\frac{1}{81}\right)$$ $$e\left(\frac{20}{81}\right)$$ $$e\left(\frac{2}{81}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{73}{81}\right)$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{40}{81}\right)$$ $$e\left(\frac{16}{81}\right)$$ $$e\left(\frac{47}{81}\right)$$
$$\chi_{163}(5,\cdot)$$ 163.h 54 Yes $$-1$$ $$1$$ $$e\left(\frac{5}{54}\right)$$ $$e\left(\frac{19}{54}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{41}{54}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{19}{54}\right)$$
$$\chi_{163}(6,\cdot)$$ 163.g 27 Yes $$1$$ $$1$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{16}{27}\right)$$
$$\chi_{163}(7,\cdot)$$ 163.j 162 Yes $$-1$$ $$1$$ $$e\left(\frac{73}{162}\right)$$ $$e\left(\frac{83}{162}\right)$$ $$e\left(\frac{73}{81}\right)$$ $$e\left(\frac{41}{54}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{145}{162}\right)$$ $$e\left(\frac{19}{54}\right)$$ $$e\left(\frac{2}{81}\right)$$ $$e\left(\frac{17}{81}\right)$$ $$e\left(\frac{29}{162}\right)$$
$$\chi_{163}(8,\cdot)$$ 163.h 54 Yes $$-1$$ $$1$$ $$e\left(\frac{1}{54}\right)$$ $$e\left(\frac{47}{54}\right)$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{19}{54}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{47}{54}\right)$$
$$\chi_{163}(9,\cdot)$$ 163.i 81 Yes $$1$$ $$1$$ $$e\left(\frac{20}{81}\right)$$ $$e\left(\frac{76}{81}\right)$$ $$e\left(\frac{40}{81}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{2}{81}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{71}{81}\right)$$ $$e\left(\frac{77}{81}\right)$$ $$e\left(\frac{49}{81}\right)$$
$$\chi_{163}(10,\cdot)$$ 163.i 81 Yes $$1$$ $$1$$ $$e\left(\frac{8}{81}\right)$$ $$e\left(\frac{79}{81}\right)$$ $$e\left(\frac{16}{81}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{17}{81}\right)$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{77}{81}\right)$$ $$e\left(\frac{47}{81}\right)$$ $$e\left(\frac{52}{81}\right)$$
$$\chi_{163}(11,\cdot)$$ 163.j 162 Yes $$-1$$ $$1$$ $$e\left(\frac{47}{162}\right)$$ $$e\left(\frac{49}{162}\right)$$ $$e\left(\frac{47}{81}\right)$$ $$e\left(\frac{19}{54}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{29}{162}\right)$$ $$e\left(\frac{47}{54}\right)$$ $$e\left(\frac{49}{81}\right)$$ $$e\left(\frac{52}{81}\right)$$ $$e\left(\frac{103}{162}\right)$$
$$\chi_{163}(12,\cdot)$$ 163.j 162 Yes $$-1$$ $$1$$ $$e\left(\frac{103}{162}\right)$$ $$e\left(\frac{35}{162}\right)$$ $$e\left(\frac{22}{81}\right)$$ $$e\left(\frac{29}{54}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{67}{162}\right)$$ $$e\left(\frac{49}{54}\right)$$ $$e\left(\frac{35}{81}\right)$$ $$e\left(\frac{14}{81}\right)$$ $$e\left(\frac{143}{162}\right)$$
$$\chi_{163}(13,\cdot)$$ 163.h 54 Yes $$-1$$ $$1$$ $$e\left(\frac{17}{54}\right)$$ $$e\left(\frac{43}{54}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{53}{54}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{43}{54}\right)$$
$$\chi_{163}(14,\cdot)$$ 163.i 81 Yes $$1$$ $$1$$ $$e\left(\frac{37}{81}\right)$$ $$e\left(\frac{11}{81}\right)$$ $$e\left(\frac{74}{81}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{28}{81}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{22}{81}\right)$$ $$e\left(\frac{25}{81}\right)$$ $$e\left(\frac{38}{81}\right)$$
$$\chi_{163}(15,\cdot)$$ 163.i 81 Yes $$1$$ $$1$$ $$e\left(\frac{58}{81}\right)$$ $$e\left(\frac{26}{81}\right)$$ $$e\left(\frac{35}{81}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{22}{81}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{52}{81}\right)$$ $$e\left(\frac{37}{81}\right)$$ $$e\left(\frac{53}{81}\right)$$
$$\chi_{163}(16,\cdot)$$ 163.i 81 Yes $$1$$ $$1$$ $$e\left(\frac{2}{81}\right)$$ $$e\left(\frac{40}{81}\right)$$ $$e\left(\frac{4}{81}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{65}{81}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{80}{81}\right)$$ $$e\left(\frac{32}{81}\right)$$ $$e\left(\frac{13}{81}\right)$$
$$\chi_{163}(17,\cdot)$$ 163.h 54 Yes $$-1$$ $$1$$ $$e\left(\frac{19}{54}\right)$$ $$e\left(\frac{29}{54}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{37}{54}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{29}{54}\right)$$
$$\chi_{163}(18,\cdot)$$ 163.j 162 Yes $$-1$$ $$1$$ $$e\left(\frac{41}{162}\right)$$ $$e\left(\frac{91}{162}\right)$$ $$e\left(\frac{41}{81}\right)$$ $$e\left(\frac{43}{54}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{77}{162}\right)$$ $$e\left(\frac{41}{54}\right)$$ $$e\left(\frac{10}{81}\right)$$ $$e\left(\frac{4}{81}\right)$$ $$e\left(\frac{145}{162}\right)$$
$$\chi_{163}(19,\cdot)$$ 163.j 162 Yes $$-1$$ $$1$$ $$e\left(\frac{125}{162}\right)$$ $$e\left(\frac{151}{162}\right)$$ $$e\left(\frac{44}{81}\right)$$ $$e\left(\frac{31}{54}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{53}{162}\right)$$ $$e\left(\frac{17}{54}\right)$$ $$e\left(\frac{70}{81}\right)$$ $$e\left(\frac{28}{81}\right)$$ $$e\left(\frac{43}{162}\right)$$
$$\chi_{163}(20,\cdot)$$ 163.j 162 Yes $$-1$$ $$1$$ $$e\left(\frac{17}{162}\right)$$ $$e\left(\frac{97}{162}\right)$$ $$e\left(\frac{17}{81}\right)$$ $$e\left(\frac{31}{54}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{107}{162}\right)$$ $$e\left(\frac{17}{54}\right)$$ $$e\left(\frac{16}{81}\right)$$ $$e\left(\frac{55}{81}\right)$$ $$e\left(\frac{151}{162}\right)$$
$$\chi_{163}(21,\cdot)$$ 163.g 27 Yes $$1$$ $$1$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{13}{27}\right)$$
$$\chi_{163}(22,\cdot)$$ 163.g 27 Yes $$1$$ $$1$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{25}{27}\right)$$
$$\chi_{163}(23,\cdot)$$ 163.f 18 Yes $$-1$$ $$1$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$
$$\chi_{163}(24,\cdot)$$ 163.i 81 Yes $$1$$ $$1$$ $$e\left(\frac{52}{81}\right)$$ $$e\left(\frac{68}{81}\right)$$ $$e\left(\frac{23}{81}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{70}{81}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{55}{81}\right)$$ $$e\left(\frac{22}{81}\right)$$ $$e\left(\frac{14}{81}\right)$$
$$\chi_{163}(25,\cdot)$$ 163.g 27 Yes $$1$$ $$1$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{19}{27}\right)$$
$$\chi_{163}(26,\cdot)$$ 163.i 81 Yes $$1$$ $$1$$ $$e\left(\frac{26}{81}\right)$$ $$e\left(\frac{34}{81}\right)$$ $$e\left(\frac{52}{81}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{35}{81}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{68}{81}\right)$$ $$e\left(\frac{11}{81}\right)$$ $$e\left(\frac{7}{81}\right)$$
$$\chi_{163}(27,\cdot)$$ 163.h 54 Yes $$-1$$ $$1$$ $$e\left(\frac{47}{54}\right)$$ $$e\left(\frac{49}{54}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{29}{54}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{49}{54}\right)$$
$$\chi_{163}(28,\cdot)$$ 163.h 54 Yes $$-1$$ $$1$$ $$e\left(\frac{25}{54}\right)$$ $$e\left(\frac{41}{54}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{43}{54}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{41}{54}\right)$$
$$\chi_{163}(29,\cdot)$$ 163.j 162 Yes $$-1$$ $$1$$ $$e\left(\frac{107}{162}\right)$$ $$e\left(\frac{115}{162}\right)$$ $$e\left(\frac{26}{81}\right)$$ $$e\left(\frac{49}{54}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{35}{162}\right)$$ $$e\left(\frac{53}{54}\right)$$ $$e\left(\frac{34}{81}\right)$$ $$e\left(\frac{46}{81}\right)$$ $$e\left(\frac{7}{162}\right)$$
$$\chi_{163}(30,\cdot)$$ 163.f 18 Yes $$-1$$ $$1$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$
$$\chi_{163}(31,\cdot)$$ 163.h 54 Yes $$-1$$ $$1$$ $$e\left(\frac{23}{54}\right)$$ $$e\left(\frac{1}{54}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{54}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{1}{54}\right)$$
$$\chi_{163}(32,\cdot)$$ 163.j 162 Yes $$-1$$ $$1$$ $$e\left(\frac{5}{162}\right)$$ $$e\left(\frac{19}{162}\right)$$ $$e\left(\frac{5}{81}\right)$$ $$e\left(\frac{25}{54}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{41}{162}\right)$$ $$e\left(\frac{5}{54}\right)$$ $$e\left(\frac{19}{81}\right)$$ $$e\left(\frac{40}{81}\right)$$ $$e\left(\frac{73}{162}\right)$$