# Properties

 Modulus $432$ Structure $$C_{2}\times C_{2}\times C_{36}$$ Order $144$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(432)

pari: g = idealstar(,432,2)

## Character group

 sage: G.order()  pari: g.no Order = 144 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{2}\times C_{36}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{432}(271,\cdot)$, $\chi_{432}(325,\cdot)$, $\chi_{432}(353,\cdot)$

## First 32 of 144 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 $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$
$$\chi_{432}(1,\cdot)$$ 432.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{432}(5,\cdot)$$ 432.bi 36 yes $$-1$$ $$1$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{5}{9}\right)$$
$$\chi_{432}(7,\cdot)$$ 432.z 18 no $$-1$$ $$1$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{5}{18}\right)$$
$$\chi_{432}(11,\cdot)$$ 432.bj 36 yes $$1$$ $$1$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{17}{18}\right)$$
$$\chi_{432}(13,\cdot)$$ 432.bg 36 yes $$1$$ $$1$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{8}{9}\right)$$
$$\chi_{432}(17,\cdot)$$ 432.q 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{432}(19,\cdot)$$ 432.w 12 no $$-1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$1$$ $$-i$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{432}(23,\cdot)$$ 432.bd 18 no $$1$$ $$1$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{13}{18}\right)$$
$$\chi_{432}(25,\cdot)$$ 432.bb 18 no $$1$$ $$1$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$
$$\chi_{432}(29,\cdot)$$ 432.bi 36 yes $$-1$$ $$1$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{1}{9}\right)$$
$$\chi_{432}(31,\cdot)$$ 432.ba 18 no $$-1$$ $$1$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{13}{18}\right)$$
$$\chi_{432}(35,\cdot)$$ 432.v 12 no $$1$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$-1$$ $$-i$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{432}(37,\cdot)$$ 432.y 12 no $$1$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$1$$ $$-i$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{432}(41,\cdot)$$ 432.bf 18 no $$-1$$ $$1$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$
$$\chi_{432}(43,\cdot)$$ 432.bh 36 yes $$-1$$ $$1$$ $$e\left(\frac{13}{36}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{17}{18}\right)$$
$$\chi_{432}(47,\cdot)$$ 432.be 18 no $$1$$ $$1$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{5}{18}\right)$$
$$\chi_{432}(49,\cdot)$$ 432.u 9 no $$1$$ $$1$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$
$$\chi_{432}(53,\cdot)$$ 432.j 4 no $$-1$$ $$1$$ $$-i$$ $$-1$$ $$-i$$ $$-i$$ $$-1$$ $$-i$$ $$1$$ $$-1$$ $$i$$ $$1$$
$$\chi_{432}(55,\cdot)$$ 432.b 2 no $$-1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$-1$$
$$\chi_{432}(59,\cdot)$$ 432.bj 36 yes $$1$$ $$1$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{13}{36}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{1}{18}\right)$$
$$\chi_{432}(61,\cdot)$$ 432.bg 36 yes $$1$$ $$1$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{13}{36}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{7}{9}\right)$$
$$\chi_{432}(65,\cdot)$$ 432.bc 18 no $$-1$$ $$1$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$
$$\chi_{432}(67,\cdot)$$ 432.bh 36 yes $$-1$$ $$1$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{7}{18}\right)$$
$$\chi_{432}(71,\cdot)$$ 432.p 6 no $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{432}(73,\cdot)$$ 432.r 6 no $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{432}(77,\cdot)$$ 432.bi 36 yes $$-1$$ $$1$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{2}{9}\right)$$
$$\chi_{432}(79,\cdot)$$ 432.ba 18 no $$-1$$ $$1$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$
$$\chi_{432}(83,\cdot)$$ 432.bj 36 yes $$1$$ $$1$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{11}{18}\right)$$
$$\chi_{432}(85,\cdot)$$ 432.bg 36 yes $$1$$ $$1$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{2}{9}\right)$$
$$\chi_{432}(89,\cdot)$$ 432.n 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{432}(91,\cdot)$$ 432.w 12 no $$-1$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$1$$ $$i$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{432}(95,\cdot)$$ 432.be 18 no $$1$$ $$1$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{7}{18}\right)$$