# Properties

 Modulus $475$ Structure $$C_{2}\times C_{180}$$ Order $360$

Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(475)

pari: g = idealstar(,475,2)

## Character group

 sage: G.order()  pari: g.no Order = 360 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{180}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{475}(77,\cdot)$, $\chi_{475}(401,\cdot)$

## First 32 of 360 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 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$6$$ $$7$$ $$8$$ $$9$$ $$11$$ $$12$$ $$13$$
$$\chi_{475}(1,\cdot)$$ 475.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{475}(2,\cdot)$$ 475.bi 180 yes $$1$$ $$1$$ $$e\left(\frac{19}{180}\right)$$ $$e\left(\frac{13}{180}\right)$$ $$e\left(\frac{19}{90}\right)$$ $$e\left(\frac{8}{45}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{19}{60}\right)$$ $$e\left(\frac{13}{90}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{17}{60}\right)$$ $$e\left(\frac{41}{180}\right)$$
$$\chi_{475}(3,\cdot)$$ 475.bi 180 yes $$1$$ $$1$$ $$e\left(\frac{13}{180}\right)$$ $$e\left(\frac{151}{180}\right)$$ $$e\left(\frac{13}{90}\right)$$ $$e\left(\frac{41}{45}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{13}{60}\right)$$ $$e\left(\frac{61}{90}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{59}{60}\right)$$ $$e\left(\frac{47}{180}\right)$$
$$\chi_{475}(4,\cdot)$$ 475.bg 90 yes $$1$$ $$1$$ $$e\left(\frac{19}{90}\right)$$ $$e\left(\frac{13}{90}\right)$$ $$e\left(\frac{19}{45}\right)$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{13}{45}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{41}{90}\right)$$
$$\chi_{475}(6,\cdot)$$ 475.bc 45 yes $$1$$ $$1$$ $$e\left(\frac{8}{45}\right)$$ $$e\left(\frac{41}{45}\right)$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{4}{45}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{37}{45}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{22}{45}\right)$$
$$\chi_{475}(7,\cdot)$$ 475.q 12 no $$-1$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$i$$ $$-i$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$i$$ $$e\left(\frac{5}{12}\right)$$
$$\chi_{475}(8,\cdot)$$ 475.be 60 yes $$1$$ $$1$$ $$e\left(\frac{19}{60}\right)$$ $$e\left(\frac{13}{60}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$-i$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{41}{60}\right)$$
$$\chi_{475}(9,\cdot)$$ 475.bg 90 yes $$1$$ $$1$$ $$e\left(\frac{13}{90}\right)$$ $$e\left(\frac{61}{90}\right)$$ $$e\left(\frac{13}{45}\right)$$ $$e\left(\frac{37}{45}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{47}{90}\right)$$
$$\chi_{475}(11,\cdot)$$ 475.r 15 yes $$1$$ $$1$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$1$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{8}{15}\right)$$
$$\chi_{475}(12,\cdot)$$ 475.be 60 yes $$1$$ $$1$$ $$e\left(\frac{17}{60}\right)$$ $$e\left(\frac{59}{60}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$i$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{43}{60}\right)$$
$$\chi_{475}(13,\cdot)$$ 475.bi 180 yes $$1$$ $$1$$ $$e\left(\frac{41}{180}\right)$$ $$e\left(\frac{47}{180}\right)$$ $$e\left(\frac{41}{90}\right)$$ $$e\left(\frac{22}{45}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{41}{60}\right)$$ $$e\left(\frac{47}{90}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{43}{60}\right)$$ $$e\left(\frac{79}{180}\right)$$
$$\chi_{475}(14,\cdot)$$ 475.bh 90 yes $$-1$$ $$1$$ $$e\left(\frac{31}{45}\right)$$ $$e\left(\frac{7}{45}\right)$$ $$e\left(\frac{17}{45}\right)$$ $$e\left(\frac{38}{45}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{14}{45}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{29}{45}\right)$$
$$\chi_{475}(16,\cdot)$$ 475.bc 45 yes $$1$$ $$1$$ $$e\left(\frac{19}{45}\right)$$ $$e\left(\frac{13}{45}\right)$$ $$e\left(\frac{38}{45}\right)$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{26}{45}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{41}{45}\right)$$
$$\chi_{475}(17,\cdot)$$ 475.bj 180 yes $$-1$$ $$1$$ $$e\left(\frac{37}{180}\right)$$ $$e\left(\frac{139}{180}\right)$$ $$e\left(\frac{37}{90}\right)$$ $$e\left(\frac{44}{45}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{37}{60}\right)$$ $$e\left(\frac{49}{90}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{11}{60}\right)$$ $$e\left(\frac{23}{180}\right)$$
$$\chi_{475}(18,\cdot)$$ 475.g 4 no $$1$$ $$1$$ $$i$$ $$-i$$ $$-1$$ $$1$$ $$-i$$ $$-i$$ $$-1$$ $$1$$ $$i$$ $$-i$$
$$\chi_{475}(21,\cdot)$$ 475.bf 90 yes $$-1$$ $$1$$ $$e\left(\frac{59}{90}\right)$$ $$e\left(\frac{83}{90}\right)$$ $$e\left(\frac{14}{45}\right)$$ $$e\left(\frac{26}{45}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{38}{45}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{61}{90}\right)$$
$$\chi_{475}(22,\cdot)$$ 475.bi 180 yes $$1$$ $$1$$ $$e\left(\frac{103}{180}\right)$$ $$e\left(\frac{61}{180}\right)$$ $$e\left(\frac{13}{90}\right)$$ $$e\left(\frac{41}{45}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{43}{60}\right)$$ $$e\left(\frac{61}{90}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{29}{60}\right)$$ $$e\left(\frac{137}{180}\right)$$
$$\chi_{475}(23,\cdot)$$ 475.bj 180 yes $$-1$$ $$1$$ $$e\left(\frac{119}{180}\right)$$ $$e\left(\frac{53}{180}\right)$$ $$e\left(\frac{29}{90}\right)$$ $$e\left(\frac{43}{45}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{59}{60}\right)$$ $$e\left(\frac{53}{90}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{37}{60}\right)$$ $$e\left(\frac{1}{180}\right)$$
$$\chi_{475}(24,\cdot)$$ 475.u 18 no $$1$$ $$1$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{17}{18}\right)$$
$$\chi_{475}(26,\cdot)$$ 475.e 3 no $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{475}(27,\cdot)$$ 475.be 60 yes $$1$$ $$1$$ $$e\left(\frac{13}{60}\right)$$ $$e\left(\frac{31}{60}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$i$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{47}{60}\right)$$
$$\chi_{475}(28,\cdot)$$ 475.bj 180 yes $$-1$$ $$1$$ $$e\left(\frac{143}{180}\right)$$ $$e\left(\frac{41}{180}\right)$$ $$e\left(\frac{53}{90}\right)$$ $$e\left(\frac{1}{45}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{23}{60}\right)$$ $$e\left(\frac{41}{90}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{49}{60}\right)$$ $$e\left(\frac{157}{180}\right)$$
$$\chi_{475}(29,\cdot)$$ 475.bh 90 yes $$-1$$ $$1$$ $$e\left(\frac{2}{45}\right)$$ $$e\left(\frac{44}{45}\right)$$ $$e\left(\frac{4}{45}\right)$$ $$e\left(\frac{1}{45}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{43}{45}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{28}{45}\right)$$
$$\chi_{475}(31,\cdot)$$ 475.y 30 yes $$-1$$ $$1$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$1$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{23}{30}\right)$$
$$\chi_{475}(32,\cdot)$$ 475.bb 36 no $$1$$ $$1$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{13}{36}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{36}\right)$$
$$\chi_{475}(33,\cdot)$$ 475.bi 180 yes $$1$$ $$1$$ $$e\left(\frac{97}{180}\right)$$ $$e\left(\frac{19}{180}\right)$$ $$e\left(\frac{7}{90}\right)$$ $$e\left(\frac{29}{45}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{37}{60}\right)$$ $$e\left(\frac{19}{90}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{11}{60}\right)$$ $$e\left(\frac{143}{180}\right)$$
$$\chi_{475}(34,\cdot)$$ 475.bh 90 yes $$-1$$ $$1$$ $$e\left(\frac{14}{45}\right)$$ $$e\left(\frac{38}{45}\right)$$ $$e\left(\frac{28}{45}\right)$$ $$e\left(\frac{7}{45}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{31}{45}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{16}{45}\right)$$
$$\chi_{475}(36,\cdot)$$ 475.bc 45 yes $$1$$ $$1$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{37}{45}\right)$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{8}{45}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{29}{45}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{44}{45}\right)$$
$$\chi_{475}(37,\cdot)$$ 475.v 20 yes $$1$$ $$1$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$i$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{1}{20}\right)$$
$$\chi_{475}(39,\cdot)$$ 475.n 10 no $$1$$ $$1$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$-1$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$
$$\chi_{475}(41,\cdot)$$ 475.bf 90 yes $$-1$$ $$1$$ $$e\left(\frac{83}{90}\right)$$ $$e\left(\frac{71}{90}\right)$$ $$e\left(\frac{38}{45}\right)$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{26}{45}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{37}{90}\right)$$
$$\chi_{475}(42,\cdot)$$ 475.bj 180 yes $$-1$$ $$1$$ $$e\left(\frac{137}{180}\right)$$ $$e\left(\frac{179}{180}\right)$$ $$e\left(\frac{47}{90}\right)$$ $$e\left(\frac{34}{45}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{17}{60}\right)$$ $$e\left(\frac{89}{90}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{31}{60}\right)$$ $$e\left(\frac{163}{180}\right)$$