# Properties

 Modulus $324$ Structure $$C_{2}\times C_{54}$$ Order $108$

Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(324)

pari: g = idealstar(,324,2)

## Character group

 sage: G.order()  pari: g.no Order = 108 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{54}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{324}(163,\cdot)$, $\chi_{324}(245,\cdot)$

## First 32 of 108 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 $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$
$$\chi_{324}(1,\cdot)$$ 324.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{324}(5,\cdot)$$ 324.o 54 no $$-1$$ $$1$$ $$e\left(\frac{43}{54}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{29}{54}\right)$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{37}{54}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{41}{54}\right)$$ $$e\left(\frac{14}{27}\right)$$
$$\chi_{324}(7,\cdot)$$ 324.n 54 yes $$-1$$ $$1$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{13}{54}\right)$$ $$e\left(\frac{19}{54}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{41}{54}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{23}{54}\right)$$
$$\chi_{324}(11,\cdot)$$ 324.p 54 yes $$1$$ $$1$$ $$e\left(\frac{29}{54}\right)$$ $$e\left(\frac{19}{54}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{49}{54}\right)$$ $$e\left(\frac{17}{54}\right)$$
$$\chi_{324}(13,\cdot)$$ 324.m 27 no $$1$$ $$1$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{26}{27}\right)$$
$$\chi_{324}(17,\cdot)$$ 324.k 18 no $$-1$$ $$1$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{2}{9}\right)$$
$$\chi_{324}(19,\cdot)$$ 324.j 18 no $$-1$$ $$1$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{5}{18}\right)$$
$$\chi_{324}(23,\cdot)$$ 324.p 54 yes $$1$$ $$1$$ $$e\left(\frac{37}{54}\right)$$ $$e\left(\frac{41}{54}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{29}{54}\right)$$ $$e\left(\frac{31}{54}\right)$$
$$\chi_{324}(25,\cdot)$$ 324.m 27 no $$1$$ $$1$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{1}{27}\right)$$
$$\chi_{324}(29,\cdot)$$ 324.o 54 no $$-1$$ $$1$$ $$e\left(\frac{41}{54}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{49}{54}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{29}{54}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{19}{54}\right)$$ $$e\left(\frac{19}{27}\right)$$
$$\chi_{324}(31,\cdot)$$ 324.n 54 yes $$-1$$ $$1$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{23}{54}\right)$$ $$e\left(\frac{17}{54}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{31}{54}\right)$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{49}{54}\right)$$
$$\chi_{324}(35,\cdot)$$ 324.l 18 no $$1$$ $$1$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{17}{18}\right)$$
$$\chi_{324}(37,\cdot)$$ 324.i 9 no $$1$$ $$1$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$
$$\chi_{324}(41,\cdot)$$ 324.o 54 no $$-1$$ $$1$$ $$e\left(\frac{31}{54}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{41}{54}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{43}{54}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{17}{54}\right)$$ $$e\left(\frac{17}{27}\right)$$
$$\chi_{324}(43,\cdot)$$ 324.n 54 yes $$-1$$ $$1$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{1}{54}\right)$$ $$e\left(\frac{43}{54}\right)$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{53}{54}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{35}{54}\right)$$
$$\chi_{324}(47,\cdot)$$ 324.p 54 yes $$1$$ $$1$$ $$e\left(\frac{53}{54}\right)$$ $$e\left(\frac{31}{54}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{43}{54}\right)$$ $$e\left(\frac{5}{54}\right)$$
$$\chi_{324}(49,\cdot)$$ 324.m 27 no $$1$$ $$1$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{23}{27}\right)$$
$$\chi_{324}(53,\cdot)$$ 324.g 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{324}(55,\cdot)$$ 324.f 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{324}(59,\cdot)$$ 324.p 54 yes $$1$$ $$1$$ $$e\left(\frac{25}{54}\right)$$ $$e\left(\frac{35}{54}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{5}{54}\right)$$ $$e\left(\frac{37}{54}\right)$$
$$\chi_{324}(61,\cdot)$$ 324.m 27 no $$1$$ $$1$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{7}{27}\right)$$
$$\chi_{324}(65,\cdot)$$ 324.o 54 no $$-1$$ $$1$$ $$e\left(\frac{11}{54}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{25}{54}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{17}{54}\right)$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{13}{54}\right)$$ $$e\left(\frac{13}{27}\right)$$
$$\chi_{324}(67,\cdot)$$ 324.n 54 yes $$-1$$ $$1$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{29}{54}\right)$$ $$e\left(\frac{5}{54}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{25}{54}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{43}{54}\right)$$
$$\chi_{324}(71,\cdot)$$ 324.l 18 no $$1$$ $$1$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{7}{18}\right)$$
$$\chi_{324}(73,\cdot)$$ 324.i 9 no $$1$$ $$1$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$
$$\chi_{324}(77,\cdot)$$ 324.o 54 no $$-1$$ $$1$$ $$e\left(\frac{19}{54}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{53}{54}\right)$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{49}{54}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{47}{54}\right)$$ $$e\left(\frac{20}{27}\right)$$
$$\chi_{324}(79,\cdot)$$ 324.n 54 yes $$-1$$ $$1$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{43}{54}\right)$$ $$e\left(\frac{13}{54}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{11}{54}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{47}{54}\right)$$
$$\chi_{324}(83,\cdot)$$ 324.p 54 yes $$1$$ $$1$$ $$e\left(\frac{23}{54}\right)$$ $$e\left(\frac{43}{54}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{37}{54}\right)$$ $$e\left(\frac{47}{54}\right)$$
$$\chi_{324}(85,\cdot)$$ 324.m 27 no $$1$$ $$1$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{20}{27}\right)$$
$$\chi_{324}(89,\cdot)$$ 324.k 18 no $$-1$$ $$1$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$
$$\chi_{324}(91,\cdot)$$ 324.j 18 no $$-1$$ $$1$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$
$$\chi_{324}(95,\cdot)$$ 324.p 54 yes $$1$$ $$1$$ $$e\left(\frac{13}{54}\right)$$ $$e\left(\frac{29}{54}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{35}{54}\right)$$ $$e\left(\frac{43}{54}\right)$$