# Properties

 Modulus $162$ Structure $$C_{54}$$ Order $54$

Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(162)

pari: g = idealstar(,162,2)

## Character group

 sage: G.order()  pari: g.no Order = 54 sage: H.invariants()  pari: g.cyc Structure = $$C_{54}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{162}(83,\cdot)$

## First 32 of 54 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 $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$
$$\chi_{162}(1,\cdot)$$ 162.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{162}(5,\cdot)$$ 162.h 54 no $$-1$$ $$1$$ $$e\left(\frac{43}{54}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{29}{54}\right)$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{37}{54}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{41}{54}\right)$$ $$e\left(\frac{14}{27}\right)$$
$$\chi_{162}(7,\cdot)$$ 162.g 27 no $$1$$ $$1$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{25}{27}\right)$$
$$\chi_{162}(11,\cdot)$$ 162.h 54 no $$-1$$ $$1$$ $$e\left(\frac{29}{54}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{7}{54}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{35}{54}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{49}{54}\right)$$ $$e\left(\frac{22}{27}\right)$$
$$\chi_{162}(13,\cdot)$$ 162.g 27 no $$1$$ $$1$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{26}{27}\right)$$
$$\chi_{162}(17,\cdot)$$ 162.f 18 no $$-1$$ $$1$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{2}{9}\right)$$
$$\chi_{162}(19,\cdot)$$ 162.e 9 no $$1$$ $$1$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$
$$\chi_{162}(23,\cdot)$$ 162.h 54 no $$-1$$ $$1$$ $$e\left(\frac{37}{54}\right)$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{35}{54}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{13}{54}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{29}{54}\right)$$ $$e\left(\frac{2}{27}\right)$$
$$\chi_{162}(25,\cdot)$$ 162.g 27 no $$1$$ $$1$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{1}{27}\right)$$
$$\chi_{162}(29,\cdot)$$ 162.h 54 no $$-1$$ $$1$$ $$e\left(\frac{41}{54}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{49}{54}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{29}{54}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{19}{54}\right)$$ $$e\left(\frac{19}{27}\right)$$
$$\chi_{162}(31,\cdot)$$ 162.g 27 no $$1$$ $$1$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{11}{27}\right)$$
$$\chi_{162}(35,\cdot)$$ 162.f 18 no $$-1$$ $$1$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$
$$\chi_{162}(37,\cdot)$$ 162.e 9 no $$1$$ $$1$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$
$$\chi_{162}(41,\cdot)$$ 162.h 54 no $$-1$$ $$1$$ $$e\left(\frac{31}{54}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{41}{54}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{43}{54}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{17}{54}\right)$$ $$e\left(\frac{17}{27}\right)$$
$$\chi_{162}(43,\cdot)$$ 162.g 27 no $$1$$ $$1$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{4}{27}\right)$$
$$\chi_{162}(47,\cdot)$$ 162.h 54 no $$-1$$ $$1$$ $$e\left(\frac{53}{54}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{37}{54}\right)$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{23}{54}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{43}{54}\right)$$ $$e\left(\frac{16}{27}\right)$$
$$\chi_{162}(49,\cdot)$$ 162.g 27 no $$1$$ $$1$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{23}{27}\right)$$
$$\chi_{162}(53,\cdot)$$ 162.d 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{162}(55,\cdot)$$ 162.c 3 no $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{162}(59,\cdot)$$ 162.h 54 no $$-1$$ $$1$$ $$e\left(\frac{25}{54}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{47}{54}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{19}{54}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{5}{54}\right)$$ $$e\left(\frac{5}{27}\right)$$
$$\chi_{162}(61,\cdot)$$ 162.g 27 no $$1$$ $$1$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{7}{27}\right)$$
$$\chi_{162}(65,\cdot)$$ 162.h 54 no $$-1$$ $$1$$ $$e\left(\frac{11}{54}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{25}{54}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{17}{54}\right)$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{13}{54}\right)$$ $$e\left(\frac{13}{27}\right)$$
$$\chi_{162}(67,\cdot)$$ 162.g 27 no $$1$$ $$1$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{8}{27}\right)$$
$$\chi_{162}(71,\cdot)$$ 162.f 18 no $$-1$$ $$1$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$
$$\chi_{162}(73,\cdot)$$ 162.e 9 no $$1$$ $$1$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$
$$\chi_{162}(77,\cdot)$$ 162.h 54 no $$-1$$ $$1$$ $$e\left(\frac{19}{54}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{53}{54}\right)$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{49}{54}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{47}{54}\right)$$ $$e\left(\frac{20}{27}\right)$$
$$\chi_{162}(79,\cdot)$$ 162.g 27 no $$1$$ $$1$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{10}{27}\right)$$
$$\chi_{162}(83,\cdot)$$ 162.h 54 no $$-1$$ $$1$$ $$e\left(\frac{23}{54}\right)$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{13}{54}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{11}{54}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{37}{54}\right)$$ $$e\left(\frac{10}{27}\right)$$
$$\chi_{162}(85,\cdot)$$ 162.g 27 no $$1$$ $$1$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{20}{27}\right)$$
$$\chi_{162}(89,\cdot)$$ 162.f 18 no $$-1$$ $$1$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$
$$\chi_{162}(91,\cdot)$$ 162.e 9 no $$1$$ $$1$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$
$$\chi_{162}(95,\cdot)$$ 162.h 54 no $$-1$$ $$1$$ $$e\left(\frac{13}{54}\right)$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{5}{54}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{25}{54}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{35}{54}\right)$$ $$e\left(\frac{8}{27}\right)$$