# Properties

 Modulus $405$ Structure $$C_{2}\times C_{108}$$ Order $216$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(405)

pari: g = idealstar(,405,2)

## Character group

 sage: G.order()  pari: g.no Order = 216 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{108}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{405}(326,\cdot)$, $\chi_{405}(82,\cdot)$

## First 32 of 216 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 $$-1$$ $$1$$ $$2$$ $$4$$ $$7$$ $$8$$ $$11$$ $$13$$ $$14$$ $$16$$ $$17$$ $$19$$
$$\chi_{405}(1,\cdot)$$ 405.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{405}(2,\cdot)$$ 405.x 108 yes $$1$$ $$1$$ $$e\left(\frac{29}{108}\right)$$ $$e\left(\frac{29}{54}\right)$$ $$e\left(\frac{59}{108}\right)$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{13}{54}\right)$$ $$e\left(\frac{97}{108}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{7}{18}\right)$$
$$\chi_{405}(4,\cdot)$$ 405.t 54 yes $$1$$ $$1$$ $$e\left(\frac{29}{54}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{5}{54}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{43}{54}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$
$$\chi_{405}(7,\cdot)$$ 405.w 108 yes $$-1$$ $$1$$ $$e\left(\frac{59}{108}\right)$$ $$e\left(\frac{5}{54}\right)$$ $$e\left(\frac{107}{108}\right)$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{13}{108}\right)$$ $$e\left(\frac{29}{54}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{13}{18}\right)$$
$$\chi_{405}(8,\cdot)$$ 405.r 36 no $$1$$ $$1$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{405}(11,\cdot)$$ 405.u 54 no $$-1$$ $$1$$ $$e\left(\frac{13}{54}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{7}{54}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{5}{54}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$
$$\chi_{405}(13,\cdot)$$ 405.w 108 yes $$-1$$ $$1$$ $$e\left(\frac{97}{108}\right)$$ $$e\left(\frac{43}{54}\right)$$ $$e\left(\frac{13}{108}\right)$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{47}{108}\right)$$ $$e\left(\frac{1}{54}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{11}{18}\right)$$
$$\chi_{405}(14,\cdot)$$ 405.v 54 yes $$-1$$ $$1$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{29}{54}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{54}\right)$$ $$e\left(\frac{1}{54}\right)$$ $$e\left(\frac{19}{54}\right)$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$
$$\chi_{405}(16,\cdot)$$ 405.q 27 no $$1$$ $$1$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$
$$\chi_{405}(17,\cdot)$$ 405.r 36 no $$1$$ $$1$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{405}(19,\cdot)$$ 405.p 18 no $$1$$ $$1$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{405}(22,\cdot)$$ 405.w 108 yes $$-1$$ $$1$$ $$e\left(\frac{55}{108}\right)$$ $$e\left(\frac{1}{54}\right)$$ $$e\left(\frac{43}{108}\right)$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{89}{108}\right)$$ $$e\left(\frac{49}{54}\right)$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{17}{18}\right)$$
$$\chi_{405}(23,\cdot)$$ 405.x 108 yes $$1$$ $$1$$ $$e\left(\frac{103}{108}\right)$$ $$e\left(\frac{49}{54}\right)$$ $$e\left(\frac{1}{108}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{35}{54}\right)$$ $$e\left(\frac{95}{108}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{5}{18}\right)$$
$$\chi_{405}(26,\cdot)$$ 405.i 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$1$$
$$\chi_{405}(28,\cdot)$$ 405.l 12 no $$-1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$i$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-i$$ $$-1$$
$$\chi_{405}(29,\cdot)$$ 405.v 54 yes $$-1$$ $$1$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{25}{54}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{49}{54}\right)$$ $$e\left(\frac{53}{54}\right)$$ $$e\left(\frac{35}{54}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$
$$\chi_{405}(31,\cdot)$$ 405.q 27 no $$1$$ $$1$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$
$$\chi_{405}(32,\cdot)$$ 405.x 108 yes $$1$$ $$1$$ $$e\left(\frac{37}{108}\right)$$ $$e\left(\frac{37}{54}\right)$$ $$e\left(\frac{79}{108}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{11}{54}\right)$$ $$e\left(\frac{53}{108}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{17}{18}\right)$$
$$\chi_{405}(34,\cdot)$$ 405.t 54 yes $$1$$ $$1$$ $$e\left(\frac{7}{54}\right)$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{31}{54}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{29}{54}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{2}{9}\right)$$
$$\chi_{405}(37,\cdot)$$ 405.s 36 no $$-1$$ $$1$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{405}(38,\cdot)$$ 405.x 108 yes $$1$$ $$1$$ $$e\left(\frac{71}{108}\right)$$ $$e\left(\frac{17}{54}\right)$$ $$e\left(\frac{29}{108}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{43}{54}\right)$$ $$e\left(\frac{55}{108}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{1}{18}\right)$$
$$\chi_{405}(41,\cdot)$$ 405.u 54 no $$-1$$ $$1$$ $$e\left(\frac{53}{54}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{41}{54}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{37}{54}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$
$$\chi_{405}(43,\cdot)$$ 405.w 108 yes $$-1$$ $$1$$ $$e\left(\frac{17}{108}\right)$$ $$e\left(\frac{17}{54}\right)$$ $$e\left(\frac{29}{108}\right)$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{55}{108}\right)$$ $$e\left(\frac{23}{54}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{1}{18}\right)$$
$$\chi_{405}(44,\cdot)$$ 405.n 18 no $$-1$$ $$1$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{405}(46,\cdot)$$ 405.k 9 no $$1$$ $$1$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{405}(47,\cdot)$$ 405.x 108 yes $$1$$ $$1$$ $$e\left(\frac{41}{108}\right)$$ $$e\left(\frac{41}{54}\right)$$ $$e\left(\frac{35}{108}\right)$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{37}{54}\right)$$ $$e\left(\frac{85}{108}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{13}{18}\right)$$
$$\chi_{405}(49,\cdot)$$ 405.t 54 yes $$1$$ $$1$$ $$e\left(\frac{5}{54}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{53}{54}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{13}{54}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$
$$\chi_{405}(52,\cdot)$$ 405.w 108 yes $$-1$$ $$1$$ $$e\left(\frac{47}{108}\right)$$ $$e\left(\frac{47}{54}\right)$$ $$e\left(\frac{23}{108}\right)$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{25}{108}\right)$$ $$e\left(\frac{35}{54}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{13}{36}\right)$$ $$e\left(\frac{7}{18}\right)$$
$$\chi_{405}(53,\cdot)$$ 405.m 12 no $$1$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$-i$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$i$$ $$-1$$
$$\chi_{405}(56,\cdot)$$ 405.u 54 no $$-1$$ $$1$$ $$e\left(\frac{19}{54}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{31}{54}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{53}{54}\right)$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$
$$\chi_{405}(58,\cdot)$$ 405.w 108 yes $$-1$$ $$1$$ $$e\left(\frac{49}{108}\right)$$ $$e\left(\frac{49}{54}\right)$$ $$e\left(\frac{1}{108}\right)$$ $$e\left(\frac{13}{36}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{95}{108}\right)$$ $$e\left(\frac{25}{54}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{5}{18}\right)$$
$$\chi_{405}(59,\cdot)$$ 405.v 54 yes $$-1$$ $$1$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{35}{54}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{47}{54}\right)$$ $$e\left(\frac{31}{54}\right)$$ $$e\left(\frac{49}{54}\right)$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$