# Properties

 Modulus $185$ Structure $$C_{4}\times C_{36}$$ Order $144$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(185)

pari: g = idealstar(,185,2)

## Character group

 sage: G.order()  pari: g.no Order = 144 sage: H.invariants()  pari: g.cyc Structure = $$C_{4}\times C_{36}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{185}(112,\cdot)$, $\chi_{185}(76,\cdot)$

## 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 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$6$$ $$7$$ $$8$$ $$9$$ $$11$$ $$12$$ $$13$$
$$\chi_{185}(1,\cdot)$$ 185.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{185}(2,\cdot)$$ 185.bc 36 yes $$1$$ $$1$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$-i$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{1}{18}\right)$$
$$\chi_{185}(3,\cdot)$$ 185.y 36 yes $$-1$$ $$1$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$-1$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{7}{36}\right)$$
$$\chi_{185}(4,\cdot)$$ 185.v 18 yes $$1$$ $$1$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$-1$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$
$$\chi_{185}(6,\cdot)$$ 185.g 4 no $$-1$$ $$1$$ $$-i$$ $$-1$$ $$-1$$ $$i$$ $$1$$ $$i$$ $$1$$ $$-1$$ $$1$$ $$i$$
$$\chi_{185}(7,\cdot)$$ 185.bd 36 yes $$-1$$ $$1$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$1$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{19}{36}\right)$$
$$\chi_{185}(8,\cdot)$$ 185.u 12 yes $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$i$$ $$e\left(\frac{5}{12}\right)$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{185}(9,\cdot)$$ 185.x 18 yes $$1$$ $$1$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$1$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{7}{18}\right)$$
$$\chi_{185}(11,\cdot)$$ 185.m 6 no $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{185}(12,\cdot)$$ 185.bd 36 yes $$-1$$ $$1$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$1$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{11}{36}\right)$$
$$\chi_{185}(13,\cdot)$$ 185.bc 36 yes $$1$$ $$1$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$i$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{11}{18}\right)$$
$$\chi_{185}(14,\cdot)$$ 185.q 12 yes $$-1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-i$$ $$e\left(\frac{5}{6}\right)$$ $$i$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$
$$\chi_{185}(16,\cdot)$$ 185.o 9 no $$1$$ $$1$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$1$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$
$$\chi_{185}(17,\cdot)$$ 185.z 36 yes $$1$$ $$1$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$i$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{8}{9}\right)$$
$$\chi_{185}(18,\cdot)$$ 185.z 36 yes $$1$$ $$1$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$-i$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{4}{9}\right)$$
$$\chi_{185}(19,\cdot)$$ 185.ba 36 yes $$-1$$ $$1$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$i$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{7}{36}\right)$$
$$\chi_{185}(21,\cdot)$$ 185.w 18 no $$1$$ $$1$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$-1$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{13}{18}\right)$$
$$\chi_{185}(22,\cdot)$$ 185.z 36 yes $$1$$ $$1$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$i$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{13}{36}\right)$$ $$e\left(\frac{2}{9}\right)$$
$$\chi_{185}(23,\cdot)$$ 185.u 12 yes $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$i$$ $$e\left(\frac{1}{12}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{185}(24,\cdot)$$ 185.ba 36 yes $$-1$$ $$1$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$-i$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{13}{36}\right)$$
$$\chi_{185}(26,\cdot)$$ 185.e 3 no $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{185}(27,\cdot)$$ 185.r 12 yes $$-1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$e\left(\frac{7}{12}\right)$$ $$i$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$
$$\chi_{185}(28,\cdot)$$ 185.y 36 yes $$-1$$ $$1$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$-1$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{23}{36}\right)$$
$$\chi_{185}(29,\cdot)$$ 185.q 12 yes $$-1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-i$$ $$e\left(\frac{1}{6}\right)$$ $$i$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$
$$\chi_{185}(31,\cdot)$$ 185.g 4 no $$-1$$ $$1$$ $$i$$ $$-1$$ $$-1$$ $$-i$$ $$1$$ $$-i$$ $$1$$ $$-1$$ $$1$$ $$-i$$
$$\chi_{185}(32,\cdot)$$ 185.bc 36 yes $$1$$ $$1$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{13}{36}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$-i$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{5}{18}\right)$$
$$\chi_{185}(33,\cdot)$$ 185.bd 36 yes $$-1$$ $$1$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$1$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{13}{36}\right)$$
$$\chi_{185}(34,\cdot)$$ 185.x 18 yes $$1$$ $$1$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$1$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{17}{18}\right)$$
$$\chi_{185}(36,\cdot)$$ 185.c 2 no $$1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$-1$$
$$\chi_{185}(38,\cdot)$$ 185.i 4 no $$-1$$ $$1$$ $$-i$$ $$i$$ $$-1$$ $$1$$ $$-i$$ $$i$$ $$-1$$ $$1$$ $$-i$$ $$i$$
$$\chi_{185}(39,\cdot)$$ 185.ba 36 yes $$-1$$ $$1$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$-i$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{29}{36}\right)$$
$$\chi_{185}(41,\cdot)$$ 185.w 18 no $$1$$ $$1$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$-1$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$