# Properties

 Modulus $285$ Structure $$C_{2}\times C_{2}\times C_{36}$$ Order $144$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(285)

pari: g = idealstar(,285,2)

## 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 = $\chi_{285}(191,\cdot)$, $\chi_{285}(172,\cdot)$, $\chi_{285}(211,\cdot)$

## 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 $$-1$$ $$1$$ $$2$$ $$4$$ $$7$$ $$8$$ $$11$$ $$13$$ $$14$$ $$16$$ $$17$$ $$22$$
$$\chi_{285}(1,\cdot)$$ 285.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{285}(2,\cdot)$$ 285.bj 36 yes $$-1$$ $$1$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{35}{36}\right)$$
$$\chi_{285}(4,\cdot)$$ 285.be 18 no $$1$$ $$1$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{17}{18}\right)$$
$$\chi_{285}(7,\cdot)$$ 285.y 12 no $$-1$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$i$$ $$-i$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$
$$\chi_{285}(8,\cdot)$$ 285.w 12 yes $$-1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-i$$ $$i$$ $$-1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$
$$\chi_{285}(11,\cdot)$$ 285.s 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$-1$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{285}(13,\cdot)$$ 285.bh 36 no $$1$$ $$1$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{13}{36}\right)$$
$$\chi_{285}(14,\cdot)$$ 285.bf 18 yes $$1$$ $$1$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$
$$\chi_{285}(16,\cdot)$$ 285.u 9 no $$1$$ $$1$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$
$$\chi_{285}(17,\cdot)$$ 285.bi 36 yes $$1$$ $$1$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{17}{36}\right)$$
$$\chi_{285}(22,\cdot)$$ 285.bh 36 no $$1$$ $$1$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{13}{36}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{23}{36}\right)$$
$$\chi_{285}(23,\cdot)$$ 285.bi 36 yes $$1$$ $$1$$ $$e\left(\frac{13}{36}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{13}{36}\right)$$ $$e\left(\frac{7}{36}\right)$$
$$\chi_{285}(26,\cdot)$$ 285.s 6 no $$-1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$-1$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{285}(28,\cdot)$$ 285.bg 36 no $$-1$$ $$1$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{19}{36}\right)$$
$$\chi_{285}(29,\cdot)$$ 285.bf 18 yes $$1$$ $$1$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$
$$\chi_{285}(31,\cdot)$$ 285.t 6 no $$-1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{285}(32,\cdot)$$ 285.bj 36 yes $$-1$$ $$1$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{31}{36}\right)$$
$$\chi_{285}(34,\cdot)$$ 285.bc 18 no $$-1$$ $$1$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$
$$\chi_{285}(37,\cdot)$$ 285.m 4 no $$1$$ $$1$$ $$-i$$ $$-1$$ $$i$$ $$i$$ $$1$$ $$i$$ $$1$$ $$1$$ $$i$$ $$-i$$
$$\chi_{285}(41,\cdot)$$ 285.z 18 no $$1$$ $$1$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{7}{18}\right)$$
$$\chi_{285}(43,\cdot)$$ 285.bg 36 no $$-1$$ $$1$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{11}{36}\right)$$
$$\chi_{285}(44,\cdot)$$ 285.bd 18 yes $$-1$$ $$1$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$
$$\chi_{285}(46,\cdot)$$ 285.t 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$-1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{285}(47,\cdot)$$ 285.bi 36 yes $$1$$ $$1$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{1}{36}\right)$$
$$\chi_{285}(49,\cdot)$$ 285.r 6 no $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$-1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{285}(52,\cdot)$$ 285.bh 36 no $$1$$ $$1$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{11}{36}\right)$$
$$\chi_{285}(53,\cdot)$$ 285.bj 36 yes $$-1$$ $$1$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{13}{36}\right)$$ $$e\left(\frac{25}{36}\right)$$
$$\chi_{285}(56,\cdot)$$ 285.h 2 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$-1$$
$$\chi_{285}(58,\cdot)$$ 285.l 4 no $$-1$$ $$1$$ $$-i$$ $$-1$$ $$-i$$ $$i$$ $$1$$ $$i$$ $$-1$$ $$1$$ $$-i$$ $$-i$$
$$\chi_{285}(59,\cdot)$$ 285.bf 18 yes $$1$$ $$1$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$
$$\chi_{285}(61,\cdot)$$ 285.u 9 no $$1$$ $$1$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$
$$\chi_{285}(62,\cdot)$$ 285.bi 36 yes $$1$$ $$1$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{29}{36}\right)$$