# Properties

 Modulus $380$ Structure $$C_{2}\times C_{2}\times C_{36}$$ Order $144$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(380)

pari: g = idealstar(,380,2)

## 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 = $\chi_{380}(191,\cdot)$, $\chi_{380}(77,\cdot)$, $\chi_{380}(21,\cdot)$

## 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 $$-1$$ $$1$$ $$3$$ $$7$$ $$9$$ $$11$$ $$13$$ $$17$$ $$21$$ $$23$$ $$27$$ $$29$$
$$\chi_{380}(1,\cdot)$$ 380.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{380}(3,\cdot)$$ 380.bi 36 yes $$-1$$ $$1$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{7}{9}\right)$$
$$\chi_{380}(7,\cdot)$$ 380.v 12 yes $$1$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$-i$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$-i$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{380}(9,\cdot)$$ 380.bd 18 no $$1$$ $$1$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{9}\right)$$
$$\chi_{380}(11,\cdot)$$ 380.q 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{380}(13,\cdot)$$ 380.bh 36 no $$1$$ $$1$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{2}{9}\right)$$
$$\chi_{380}(17,\cdot)$$ 380.bg 36 no $$-1$$ $$1$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{17}{18}\right)$$
$$\chi_{380}(21,\cdot)$$ 380.z 18 no $$-1$$ $$1$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{17}{18}\right)$$
$$\chi_{380}(23,\cdot)$$ 380.bj 36 yes $$1$$ $$1$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{7}{18}\right)$$
$$\chi_{380}(27,\cdot)$$ 380.w 12 yes $$-1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$-i$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$i$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{380}(29,\cdot)$$ 380.bc 18 no $$-1$$ $$1$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{18}\right)$$
$$\chi_{380}(31,\cdot)$$ 380.n 6 no $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{380}(33,\cdot)$$ 380.bh 36 no $$1$$ $$1$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{9}\right)$$
$$\chi_{380}(37,\cdot)$$ 380.l 4 no $$1$$ $$1$$ $$i$$ $$i$$ $$-1$$ $$1$$ $$i$$ $$i$$ $$-1$$ $$-i$$ $$-i$$ $$1$$
$$\chi_{380}(39,\cdot)$$ 380.h 2 no $$-1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{380}(41,\cdot)$$ 380.z 18 no $$-1$$ $$1$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{18}\right)$$
$$\chi_{380}(43,\cdot)$$ 380.bj 36 yes $$1$$ $$1$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{11}{18}\right)$$
$$\chi_{380}(47,\cdot)$$ 380.bj 36 yes $$1$$ $$1$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{18}\right)$$
$$\chi_{380}(49,\cdot)$$ 380.r 6 no $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{380}(51,\cdot)$$ 380.be 18 no $$1$$ $$1$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{13}{18}\right)$$
$$\chi_{380}(53,\cdot)$$ 380.bh 36 no $$1$$ $$1$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{8}{9}\right)$$
$$\chi_{380}(59,\cdot)$$ 380.bb 18 yes $$1$$ $$1$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{17}{18}\right)$$
$$\chi_{380}(61,\cdot)$$ 380.u 9 no $$1$$ $$1$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{8}{9}\right)$$
$$\chi_{380}(63,\cdot)$$ 380.bj 36 yes $$1$$ $$1$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{13}{18}\right)$$
$$\chi_{380}(67,\cdot)$$ 380.bi 36 yes $$-1$$ $$1$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{5}{9}\right)$$
$$\chi_{380}(69,\cdot)$$ 380.o 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{380}(71,\cdot)$$ 380.be 18 no $$1$$ $$1$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{11}{18}\right)$$
$$\chi_{380}(73,\cdot)$$ 380.bg 36 no $$-1$$ $$1$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{13}{36}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{18}\right)$$
$$\chi_{380}(77,\cdot)$$ 380.m 4 no $$-1$$ $$1$$ $$-i$$ $$i$$ $$-1$$ $$1$$ $$-i$$ $$i$$ $$1$$ $$-i$$ $$i$$ $$-1$$
$$\chi_{380}(79,\cdot)$$ 380.bb 18 yes $$1$$ $$1$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{18}\right)$$
$$\chi_{380}(81,\cdot)$$ 380.u 9 no $$1$$ $$1$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{9}\right)$$
$$\chi_{380}(83,\cdot)$$ 380.v 12 yes $$1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$i$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$i$$ $$e\left(\frac{1}{6}\right)$$