# Properties

 Modulus $247$ Structure $$C_{6}\times C_{36}$$ Order $216$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(247)

pari: g = idealstar(,247,2)

## Character group

 sage: G.order()  pari: g.no Order = 216 sage: H.invariants()  pari: g.cyc Structure = $$C_{6}\times C_{36}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{247}(210,\cdot)$, $\chi_{247}(40,\cdot)$

## 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 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$
$$\chi_{247}(1,\cdot)$$ 247.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{247}(2,\cdot)$$ 247.br 36 yes $$1$$ $$1$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{7}{36}\right)$$ $$i$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$i$$
$$\chi_{247}(3,\cdot)$$ 247.bp 18 yes $$-1$$ $$1$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$1$$
$$\chi_{247}(4,\cdot)$$ 247.bi 18 yes $$1$$ $$1$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$-1$$
$$\chi_{247}(5,\cdot)$$ 247.bv 36 yes $$-1$$ $$1$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{11}{12}\right)$$
$$\chi_{247}(6,\cdot)$$ 247.bu 36 yes $$-1$$ $$1$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$i$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$i$$
$$\chi_{247}(7,\cdot)$$ 247.bg 12 yes $$-1$$ $$1$$ $$i$$ $$1$$ $$-1$$ $$e\left(\frac{7}{12}\right)$$ $$i$$ $$e\left(\frac{1}{12}\right)$$ $$-i$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$
$$\chi_{247}(8,\cdot)$$ 247.be 12 yes $$1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$-i$$ $$i$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-i$$
$$\chi_{247}(9,\cdot)$$ 247.y 9 yes $$1$$ $$1$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$1$$
$$\chi_{247}(10,\cdot)$$ 247.bm 18 yes $$-1$$ $$1$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{247}(11,\cdot)$$ 247.bg 12 yes $$-1$$ $$1$$ $$i$$ $$1$$ $$-1$$ $$e\left(\frac{11}{12}\right)$$ $$i$$ $$e\left(\frac{5}{12}\right)$$ $$-i$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$
$$\chi_{247}(12,\cdot)$$ 247.n 6 yes $$-1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$
$$\chi_{247}(14,\cdot)$$ 247.bk 18 no $$-1$$ $$1$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{247}(15,\cdot)$$ 247.bq 36 yes $$1$$ $$1$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{11}{12}\right)$$
$$\chi_{247}(16,\cdot)$$ 247.y 9 yes $$1$$ $$1$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$1$$
$$\chi_{247}(17,\cdot)$$ 247.bo 18 yes $$1$$ $$1$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{247}(18,\cdot)$$ 247.i 4 yes $$1$$ $$1$$ $$i$$ $$-1$$ $$-1$$ $$-i$$ $$-i$$ $$i$$ $$-i$$ $$1$$ $$1$$ $$i$$
$$\chi_{247}(20,\cdot)$$ 247.bc 12 no $$-1$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$i$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$-i$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$
$$\chi_{247}(21,\cdot)$$ 247.bs 36 yes $$1$$ $$1$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{12}\right)$$
$$\chi_{247}(22,\cdot)$$ 247.bj 18 yes $$-1$$ $$1$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{247}(23,\cdot)$$ 247.bo 18 yes $$1$$ $$1$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{247}(24,\cdot)$$ 247.bu 36 yes $$-1$$ $$1$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{13}{36}\right)$$ $$-i$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$-i$$
$$\chi_{247}(25,\cdot)$$ 247.bn 18 yes $$1$$ $$1$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{247}(27,\cdot)$$ 247.t 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$1$$
$$\chi_{247}(28,\cdot)$$ 247.bt 36 yes $$-1$$ $$1$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{11}{12}\right)$$
$$\chi_{247}(29,\cdot)$$ 247.bj 18 yes $$-1$$ $$1$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{247}(30,\cdot)$$ 247.r 6 yes $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{247}(31,\cdot)$$ 247.be 12 yes $$1$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$i$$ $$-i$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$i$$
$$\chi_{247}(32,\cdot)$$ 247.br 36 yes $$1$$ $$1$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$i$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$i$$
$$\chi_{247}(33,\cdot)$$ 247.bq 36 yes $$1$$ $$1$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{12}\right)$$
$$\chi_{247}(34,\cdot)$$ 247.bs 36 yes $$1$$ $$1$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{12}\right)$$
$$\chi_{247}(35,\cdot)$$ 247.w 9 yes $$1$$ $$1$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$