# Properties

 Modulus $407$ Structure $$C_{180}\times C_{2}$$ Order $360$

Show commands for: Pari/GP / SageMath

sage: H = DirichletGroup(407)

pari: g = idealstar(,407,2)

## Character group

 sage: G.order()  pari: g.no Order = 360 sage: H.invariants()  pari: g.cyc Structure = $$C_{180}\times C_{2}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{407}(112,\cdot)$, $\chi_{407}(298,\cdot)$

## 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 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$12$$
$$\chi_{407}(1,\cdot)$$ 407.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{407}(2,\cdot)$$ 407.bj 180 yes $$1$$ $$1$$ $$e\left(\frac{23}{180}\right)$$ $$e\left(\frac{47}{90}\right)$$ $$e\left(\frac{23}{90}\right)$$ $$e\left(\frac{7}{180}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{53}{90}\right)$$ $$e\left(\frac{23}{60}\right)$$ $$e\left(\frac{2}{45}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{9}\right)$$
$$\chi_{407}(3,\cdot)$$ 407.bg 90 yes $$1$$ $$1$$ $$e\left(\frac{47}{90}\right)$$ $$e\left(\frac{8}{45}\right)$$ $$e\left(\frac{2}{45}\right)$$ $$e\left(\frac{73}{90}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{9}\right)$$
$$\chi_{407}(4,\cdot)$$ 407.bg 90 yes $$1$$ $$1$$ $$e\left(\frac{23}{90}\right)$$ $$e\left(\frac{2}{45}\right)$$ $$e\left(\frac{23}{45}\right)$$ $$e\left(\frac{7}{90}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{8}{45}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{4}{45}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{9}\right)$$
$$\chi_{407}(5,\cdot)$$ 407.bi 180 yes $$-1$$ $$1$$ $$e\left(\frac{7}{180}\right)$$ $$e\left(\frac{73}{90}\right)$$ $$e\left(\frac{7}{90}\right)$$ $$e\left(\frac{53}{180}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{11}{45}\right)$$ $$e\left(\frac{7}{60}\right)$$ $$e\left(\frac{28}{45}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{8}{9}\right)$$
$$\chi_{407}(6,\cdot)$$ 407.w 20 yes $$1$$ $$1$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$-1$$ $$1$$
$$\chi_{407}(7,\cdot)$$ 407.bh 90 yes $$-1$$ $$1$$ $$e\left(\frac{53}{90}\right)$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{8}{45}\right)$$ $$e\left(\frac{11}{45}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{31}{90}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{19}{45}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{8}{9}\right)$$
$$\chi_{407}(8,\cdot)$$ 407.bd 60 yes $$1$$ $$1$$ $$e\left(\frac{23}{60}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{7}{60}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{407}(9,\cdot)$$ 407.bc 45 yes $$1$$ $$1$$ $$e\left(\frac{2}{45}\right)$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{4}{45}\right)$$ $$e\left(\frac{28}{45}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{19}{45}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{4}{9}\right)$$
$$\chi_{407}(10,\cdot)$$ 407.j 6 yes $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{407}(12,\cdot)$$ 407.l 9 no $$1$$ $$1$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$1$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{7}{9}\right)$$
$$\chi_{407}(13,\cdot)$$ 407.bj 180 yes $$1$$ $$1$$ $$e\left(\frac{73}{180}\right)$$ $$e\left(\frac{67}{90}\right)$$ $$e\left(\frac{73}{90}\right)$$ $$e\left(\frac{77}{180}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{43}{90}\right)$$ $$e\left(\frac{13}{60}\right)$$ $$e\left(\frac{22}{45}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{9}\right)$$
$$\chi_{407}(14,\cdot)$$ 407.be 60 yes $$-1$$ $$1$$ $$e\left(\frac{43}{60}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{17}{60}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{407}(15,\cdot)$$ 407.bi 180 yes $$-1$$ $$1$$ $$e\left(\frac{101}{180}\right)$$ $$e\left(\frac{89}{90}\right)$$ $$e\left(\frac{11}{90}\right)$$ $$e\left(\frac{19}{180}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{43}{45}\right)$$ $$e\left(\frac{41}{60}\right)$$ $$e\left(\frac{44}{45}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{9}\right)$$
$$\chi_{407}(16,\cdot)$$ 407.bc 45 yes $$1$$ $$1$$ $$e\left(\frac{23}{45}\right)$$ $$e\left(\frac{4}{45}\right)$$ $$e\left(\frac{1}{45}\right)$$ $$e\left(\frac{7}{45}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{8}{45}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{9}\right)$$
$$\chi_{407}(17,\cdot)$$ 407.bj 180 yes $$1$$ $$1$$ $$e\left(\frac{17}{180}\right)$$ $$e\left(\frac{23}{90}\right)$$ $$e\left(\frac{17}{90}\right)$$ $$e\left(\frac{13}{180}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{47}{90}\right)$$ $$e\left(\frac{17}{60}\right)$$ $$e\left(\frac{23}{45}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{4}{9}\right)$$
$$\chi_{407}(18,\cdot)$$ 407.bj 180 yes $$1$$ $$1$$ $$e\left(\frac{31}{180}\right)$$ $$e\left(\frac{79}{90}\right)$$ $$e\left(\frac{31}{90}\right)$$ $$e\left(\frac{119}{180}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{1}{90}\right)$$ $$e\left(\frac{31}{60}\right)$$ $$e\left(\frac{34}{45}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{9}\right)$$
$$\chi_{407}(19,\cdot)$$ 407.bj 180 yes $$1$$ $$1$$ $$e\left(\frac{49}{180}\right)$$ $$e\left(\frac{61}{90}\right)$$ $$e\left(\frac{49}{90}\right)$$ $$e\left(\frac{101}{180}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{19}{90}\right)$$ $$e\left(\frac{49}{60}\right)$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{9}\right)$$
$$\chi_{407}(20,\cdot)$$ 407.bi 180 yes $$-1$$ $$1$$ $$e\left(\frac{53}{180}\right)$$ $$e\left(\frac{77}{90}\right)$$ $$e\left(\frac{53}{90}\right)$$ $$e\left(\frac{67}{180}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{19}{45}\right)$$ $$e\left(\frac{53}{60}\right)$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{4}{9}\right)$$
$$\chi_{407}(21,\cdot)$$ 407.u 18 yes $$-1$$ $$1$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$1$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{9}\right)$$
$$\chi_{407}(23,\cdot)$$ 407.p 12 no $$-1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$i$$ $$e\left(\frac{1}{3}\right)$$ $$i$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{407}(24,\cdot)$$ 407.bj 180 yes $$1$$ $$1$$ $$e\left(\frac{163}{180}\right)$$ $$e\left(\frac{67}{90}\right)$$ $$e\left(\frac{73}{90}\right)$$ $$e\left(\frac{167}{180}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{43}{90}\right)$$ $$e\left(\frac{43}{60}\right)$$ $$e\left(\frac{22}{45}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{9}\right)$$
$$\chi_{407}(25,\cdot)$$ 407.bg 90 yes $$1$$ $$1$$ $$e\left(\frac{7}{90}\right)$$ $$e\left(\frac{28}{45}\right)$$ $$e\left(\frac{7}{45}\right)$$ $$e\left(\frac{53}{90}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{22}{45}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{11}{45}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{7}{9}\right)$$
$$\chi_{407}(26,\cdot)$$ 407.r 15 yes $$1$$ $$1$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{407}(27,\cdot)$$ 407.x 30 yes $$1$$ $$1$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{407}(28,\cdot)$$ 407.bf 90 yes $$-1$$ $$1$$ $$e\left(\frac{38}{45}\right)$$ $$e\left(\frac{34}{45}\right)$$ $$e\left(\frac{31}{45}\right)$$ $$e\left(\frac{29}{90}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{47}{90}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{23}{45}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{4}{9}\right)$$
$$\chi_{407}(29,\cdot)$$ 407.bd 60 yes $$1$$ $$1$$ $$e\left(\frac{17}{60}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{13}{60}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{407}(30,\cdot)$$ 407.bf 90 yes $$-1$$ $$1$$ $$e\left(\frac{31}{45}\right)$$ $$e\left(\frac{23}{45}\right)$$ $$e\left(\frac{17}{45}\right)$$ $$e\left(\frac{13}{90}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{49}{90}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{1}{45}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{8}{9}\right)$$
$$\chi_{407}(31,\cdot)$$ 407.v 20 yes $$-1$$ $$1$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$1$$ $$1$$
$$\chi_{407}(32,\cdot)$$ 407.bb 36 yes $$1$$ $$1$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{7}{36}\right)$$ $$i$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{8}{9}\right)$$
$$\chi_{407}(34,\cdot)$$ 407.l 9 no $$1$$ $$1$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$1$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{9}\right)$$
$$\chi_{407}(35,\cdot)$$ 407.bj 180 yes $$1$$ $$1$$ $$e\left(\frac{113}{180}\right)$$ $$e\left(\frac{47}{90}\right)$$ $$e\left(\frac{23}{90}\right)$$ $$e\left(\frac{97}{180}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{53}{90}\right)$$ $$e\left(\frac{53}{60}\right)$$ $$e\left(\frac{2}{45}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{9}\right)$$