# Properties

 Modulus $297$ Structure $$C_{2}\times C_{90}$$ Order $180$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(297)

pari: g = idealstar(,297,2)

## Character group

 sage: G.order()  pari: g.no Order = 180 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{90}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{297}(56,\cdot)$, $\chi_{297}(244,\cdot)$

## First 32 of 180 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 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$13$$ $$14$$ $$16$$ $$17$$
$$\chi_{297}(1,\cdot)$$ 297.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{297}(2,\cdot)$$ 297.x 90 yes $$1$$ $$1$$ $$e\left(\frac{7}{45}\right)$$ $$e\left(\frac{14}{45}\right)$$ $$e\left(\frac{61}{90}\right)$$ $$e\left(\frac{53}{90}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{49}{90}\right)$$ $$e\left(\frac{67}{90}\right)$$ $$e\left(\frac{28}{45}\right)$$ $$e\left(\frac{11}{15}\right)$$
$$\chi_{297}(4,\cdot)$$ 297.u 45 yes $$1$$ $$1$$ $$e\left(\frac{14}{45}\right)$$ $$e\left(\frac{28}{45}\right)$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{8}{45}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{4}{45}\right)$$ $$e\left(\frac{22}{45}\right)$$ $$e\left(\frac{11}{45}\right)$$ $$e\left(\frac{7}{15}\right)$$
$$\chi_{297}(5,\cdot)$$ 297.v 90 yes $$-1$$ $$1$$ $$e\left(\frac{61}{90}\right)$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{89}{90}\right)$$ $$e\left(\frac{11}{45}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{28}{45}\right)$$ $$e\left(\frac{83}{90}\right)$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{23}{30}\right)$$
$$\chi_{297}(7,\cdot)$$ 297.w 90 yes $$-1$$ $$1$$ $$e\left(\frac{53}{90}\right)$$ $$e\left(\frac{8}{45}\right)$$ $$e\left(\frac{11}{45}\right)$$ $$e\left(\frac{11}{90}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{73}{90}\right)$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{19}{30}\right)$$
$$\chi_{297}(8,\cdot)$$ 297.t 30 no $$1$$ $$1$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$-1$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$
$$\chi_{297}(10,\cdot)$$ 297.h 6 no $$-1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$
$$\chi_{297}(13,\cdot)$$ 297.w 90 yes $$-1$$ $$1$$ $$e\left(\frac{49}{90}\right)$$ $$e\left(\frac{4}{45}\right)$$ $$e\left(\frac{28}{45}\right)$$ $$e\left(\frac{73}{90}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{59}{90}\right)$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{8}{45}\right)$$ $$e\left(\frac{17}{30}\right)$$
$$\chi_{297}(14,\cdot)$$ 297.v 90 yes $$-1$$ $$1$$ $$e\left(\frac{67}{90}\right)$$ $$e\left(\frac{22}{45}\right)$$ $$e\left(\frac{83}{90}\right)$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{41}{90}\right)$$ $$e\left(\frac{44}{45}\right)$$ $$e\left(\frac{11}{30}\right)$$
$$\chi_{297}(16,\cdot)$$ 297.u 45 yes $$1$$ $$1$$ $$e\left(\frac{28}{45}\right)$$ $$e\left(\frac{11}{45}\right)$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{8}{45}\right)$$ $$e\left(\frac{44}{45}\right)$$ $$e\left(\frac{22}{45}\right)$$ $$e\left(\frac{14}{15}\right)$$
$$\chi_{297}(17,\cdot)$$ 297.t 30 no $$1$$ $$1$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$-1$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$
$$\chi_{297}(19,\cdot)$$ 297.s 30 no $$-1$$ $$1$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$-1$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{7}{10}\right)$$
$$\chi_{297}(20,\cdot)$$ 297.v 90 yes $$-1$$ $$1$$ $$e\left(\frac{89}{90}\right)$$ $$e\left(\frac{44}{45}\right)$$ $$e\left(\frac{31}{90}\right)$$ $$e\left(\frac{19}{45}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{37}{90}\right)$$ $$e\left(\frac{43}{45}\right)$$ $$e\left(\frac{7}{30}\right)$$
$$\chi_{297}(23,\cdot)$$ 297.p 18 no $$-1$$ $$1$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{297}(25,\cdot)$$ 297.u 45 yes $$1$$ $$1$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{44}{45}\right)$$ $$e\left(\frac{22}{45}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{11}{45}\right)$$ $$e\left(\frac{38}{45}\right)$$ $$e\left(\frac{19}{45}\right)$$ $$e\left(\frac{8}{15}\right)$$
$$\chi_{297}(26,\cdot)$$ 297.m 10 no $$-1$$ $$1$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{297}(28,\cdot)$$ 297.l 10 no $$-1$$ $$1$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$-1$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$
$$\chi_{297}(29,\cdot)$$ 297.x 90 yes $$1$$ $$1$$ $$e\left(\frac{34}{45}\right)$$ $$e\left(\frac{23}{45}\right)$$ $$e\left(\frac{7}{90}\right)$$ $$e\left(\frac{71}{90}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{13}{90}\right)$$ $$e\left(\frac{49}{90}\right)$$ $$e\left(\frac{1}{45}\right)$$ $$e\left(\frac{2}{15}\right)$$
$$\chi_{297}(31,\cdot)$$ 297.u 45 yes $$1$$ $$1$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{19}{45}\right)$$ $$e\left(\frac{43}{45}\right)$$ $$e\left(\frac{44}{45}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{22}{45}\right)$$ $$e\left(\frac{31}{45}\right)$$ $$e\left(\frac{38}{45}\right)$$ $$e\left(\frac{1}{15}\right)$$
$$\chi_{297}(32,\cdot)$$ 297.o 18 yes $$1$$ $$1$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{297}(34,\cdot)$$ 297.j 9 no $$1$$ $$1$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{297}(35,\cdot)$$ 297.t 30 no $$1$$ $$1$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$-1$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$
$$\chi_{297}(37,\cdot)$$ 297.n 15 no $$1$$ $$1$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$1$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{4}{5}\right)$$
$$\chi_{297}(38,\cdot)$$ 297.v 90 yes $$-1$$ $$1$$ $$e\left(\frac{11}{90}\right)$$ $$e\left(\frac{11}{45}\right)$$ $$e\left(\frac{19}{90}\right)$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{8}{45}\right)$$ $$e\left(\frac{43}{90}\right)$$ $$e\left(\frac{22}{45}\right)$$ $$e\left(\frac{13}{30}\right)$$
$$\chi_{297}(40,\cdot)$$ 297.w 90 yes $$-1$$ $$1$$ $$e\left(\frac{13}{90}\right)$$ $$e\left(\frac{13}{45}\right)$$ $$e\left(\frac{1}{45}\right)$$ $$e\left(\frac{1}{90}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{23}{90}\right)$$ $$e\left(\frac{7}{45}\right)$$ $$e\left(\frac{26}{45}\right)$$ $$e\left(\frac{29}{30}\right)$$
$$\chi_{297}(41,\cdot)$$ 297.x 90 yes $$1$$ $$1$$ $$e\left(\frac{11}{45}\right)$$ $$e\left(\frac{22}{45}\right)$$ $$e\left(\frac{83}{90}\right)$$ $$e\left(\frac{19}{90}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{77}{90}\right)$$ $$e\left(\frac{41}{90}\right)$$ $$e\left(\frac{44}{45}\right)$$ $$e\left(\frac{13}{15}\right)$$
$$\chi_{297}(43,\cdot)$$ 297.q 18 yes $$-1$$ $$1$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{297}(46,\cdot)$$ 297.s 30 no $$-1$$ $$1$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$-1$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{9}{10}\right)$$
$$\chi_{297}(47,\cdot)$$ 297.v 90 yes $$-1$$ $$1$$ $$e\left(\frac{17}{90}\right)$$ $$e\left(\frac{17}{45}\right)$$ $$e\left(\frac{13}{90}\right)$$ $$e\left(\frac{37}{45}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{41}{45}\right)$$ $$e\left(\frac{1}{90}\right)$$ $$e\left(\frac{34}{45}\right)$$ $$e\left(\frac{1}{30}\right)$$
$$\chi_{297}(49,\cdot)$$ 297.u 45 yes $$1$$ $$1$$ $$e\left(\frac{8}{45}\right)$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{22}{45}\right)$$ $$e\left(\frac{11}{45}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{28}{45}\right)$$ $$e\left(\frac{19}{45}\right)$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{4}{15}\right)$$
$$\chi_{297}(50,\cdot)$$ 297.x 90 yes $$1$$ $$1$$ $$e\left(\frac{23}{45}\right)$$ $$e\left(\frac{1}{45}\right)$$ $$e\left(\frac{59}{90}\right)$$ $$e\left(\frac{7}{90}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{71}{90}\right)$$ $$e\left(\frac{53}{90}\right)$$ $$e\left(\frac{2}{45}\right)$$ $$e\left(\frac{4}{15}\right)$$
$$\chi_{297}(52,\cdot)$$ 297.w 90 yes $$-1$$ $$1$$ $$e\left(\frac{77}{90}\right)$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{44}{45}\right)$$ $$e\left(\frac{89}{90}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{67}{90}\right)$$ $$e\left(\frac{38}{45}\right)$$ $$e\left(\frac{19}{45}\right)$$ $$e\left(\frac{1}{30}\right)$$