# Properties

 Modulus $81$ Structure $$C_{54}$$ Order $54$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(81)

pari: g = idealstar(,81,2)

## 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 = $\chi_{81}(2,\cdot)$

## 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 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$13$$ $$14$$ $$16$$
$$\chi_{81}(1,\cdot)$$ 81.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{81}(2,\cdot)$$ 81.h 54 yes $$-1$$ $$1$$ $$e\left(\frac{1}{54}\right)$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{23}{54}\right)$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{13}{54}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{17}{54}\right)$$ $$e\left(\frac{2}{27}\right)$$
$$\chi_{81}(4,\cdot)$$ 81.g 27 yes $$1$$ $$1$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{4}{27}\right)$$
$$\chi_{81}(5,\cdot)$$ 81.h 54 yes $$-1$$ $$1$$ $$e\left(\frac{23}{54}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{43}{54}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{29}{54}\right)$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{13}{54}\right)$$ $$e\left(\frac{19}{27}\right)$$
$$\chi_{81}(7,\cdot)$$ 81.g 27 yes $$1$$ $$1$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{5}{27}\right)$$
$$\chi_{81}(8,\cdot)$$ 81.f 18 no $$-1$$ $$1$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{2}{9}\right)$$
$$\chi_{81}(10,\cdot)$$ 81.e 9 no $$1$$ $$1$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$
$$\chi_{81}(11,\cdot)$$ 81.h 54 yes $$-1$$ $$1$$ $$e\left(\frac{13}{54}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{29}{54}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{7}{54}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{5}{54}\right)$$ $$e\left(\frac{26}{27}\right)$$
$$\chi_{81}(13,\cdot)$$ 81.g 27 yes $$1$$ $$1$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{16}{27}\right)$$
$$\chi_{81}(14,\cdot)$$ 81.h 54 yes $$-1$$ $$1$$ $$e\left(\frac{17}{54}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{13}{54}\right)$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{5}{54}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{19}{54}\right)$$ $$e\left(\frac{7}{27}\right)$$
$$\chi_{81}(16,\cdot)$$ 81.g 27 yes $$1$$ $$1$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{8}{27}\right)$$
$$\chi_{81}(17,\cdot)$$ 81.f 18 no $$-1$$ $$1$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$
$$\chi_{81}(19,\cdot)$$ 81.e 9 no $$1$$ $$1$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$
$$\chi_{81}(20,\cdot)$$ 81.h 54 yes $$-1$$ $$1$$ $$e\left(\frac{25}{54}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{35}{54}\right)$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{1}{54}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{47}{54}\right)$$ $$e\left(\frac{23}{27}\right)$$
$$\chi_{81}(22,\cdot)$$ 81.g 27 yes $$1$$ $$1$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{1}{27}\right)$$
$$\chi_{81}(23,\cdot)$$ 81.h 54 yes $$-1$$ $$1$$ $$e\left(\frac{11}{54}\right)$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{37}{54}\right)$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{35}{54}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{25}{54}\right)$$ $$e\left(\frac{22}{27}\right)$$
$$\chi_{81}(25,\cdot)$$ 81.g 27 yes $$1$$ $$1$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{11}{27}\right)$$
$$\chi_{81}(26,\cdot)$$ 81.d 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{81}(28,\cdot)$$ 81.c 3 no $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{81}(29,\cdot)$$ 81.h 54 yes $$-1$$ $$1$$ $$e\left(\frac{37}{54}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{41}{54}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{49}{54}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{35}{54}\right)$$ $$e\left(\frac{20}{27}\right)$$
$$\chi_{81}(31,\cdot)$$ 81.g 27 yes $$1$$ $$1$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{13}{27}\right)$$
$$\chi_{81}(32,\cdot)$$ 81.h 54 yes $$-1$$ $$1$$ $$e\left(\frac{5}{54}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{7}{54}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{11}{54}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{31}{54}\right)$$ $$e\left(\frac{10}{27}\right)$$
$$\chi_{81}(34,\cdot)$$ 81.g 27 yes $$1$$ $$1$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{14}{27}\right)$$
$$\chi_{81}(35,\cdot)$$ 81.f 18 no $$-1$$ $$1$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$
$$\chi_{81}(37,\cdot)$$ 81.e 9 no $$1$$ $$1$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$
$$\chi_{81}(38,\cdot)$$ 81.h 54 yes $$-1$$ $$1$$ $$e\left(\frac{49}{54}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{47}{54}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{43}{54}\right)$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{23}{54}\right)$$ $$e\left(\frac{17}{27}\right)$$
$$\chi_{81}(40,\cdot)$$ 81.g 27 yes $$1$$ $$1$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{25}{27}\right)$$
$$\chi_{81}(41,\cdot)$$ 81.h 54 yes $$-1$$ $$1$$ $$e\left(\frac{53}{54}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{31}{54}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{41}{54}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{37}{54}\right)$$ $$e\left(\frac{25}{27}\right)$$
$$\chi_{81}(43,\cdot)$$ 81.g 27 yes $$1$$ $$1$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{17}{27}\right)$$
$$\chi_{81}(44,\cdot)$$ 81.f 18 no $$-1$$ $$1$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$
$$\chi_{81}(46,\cdot)$$ 81.e 9 no $$1$$ $$1$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$
$$\chi_{81}(47,\cdot)$$ 81.h 54 yes $$-1$$ $$1$$ $$e\left(\frac{7}{54}\right)$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{53}{54}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{37}{54}\right)$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{11}{54}\right)$$ $$e\left(\frac{14}{27}\right)$$