# Properties

 Modulus $639$ Structure $$C_{2}\times C_{210}$$ Order $420$

Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(639)

pari: g = idealstar(,639,2)

## Character group

 sage: G.order()  pari: g.no Order = 420 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{210}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{639}(569,\cdot)$, $\chi_{639}(433,\cdot)$

## First 32 of 420 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 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$13$$ $$14$$ $$16$$
$$\chi_{639}(1,\cdot)$$ 639.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{639}(2,\cdot)$$ 639.be 210 yes $$-1$$ $$1$$ $$e\left(\frac{143}{210}\right)$$ $$e\left(\frac{38}{105}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{79}{105}\right)$$ $$e\left(\frac{3}{70}\right)$$ $$e\left(\frac{32}{35}\right)$$ $$e\left(\frac{173}{210}\right)$$ $$e\left(\frac{71}{105}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{76}{105}\right)$$
$$\chi_{639}(4,\cdot)$$ 639.bc 105 yes $$1$$ $$1$$ $$e\left(\frac{38}{105}\right)$$ $$e\left(\frac{76}{105}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{53}{105}\right)$$ $$e\left(\frac{3}{35}\right)$$ $$e\left(\frac{29}{35}\right)$$ $$e\left(\frac{68}{105}\right)$$ $$e\left(\frac{37}{105}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{47}{105}\right)$$
$$\chi_{639}(5,\cdot)$$ 639.t 30 yes $$-1$$ $$1$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{14}{15}\right)$$
$$\chi_{639}(7,\cdot)$$ 639.bd 210 yes $$-1$$ $$1$$ $$e\left(\frac{79}{105}\right)$$ $$e\left(\frac{53}{105}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{143}{210}\right)$$ $$e\left(\frac{9}{35}\right)$$ $$e\left(\frac{17}{35}\right)$$ $$e\left(\frac{23}{210}\right)$$ $$e\left(\frac{187}{210}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{1}{105}\right)$$
$$\chi_{639}(8,\cdot)$$ 639.ba 70 no $$-1$$ $$1$$ $$e\left(\frac{3}{70}\right)$$ $$e\left(\frac{3}{35}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{9}{35}\right)$$ $$e\left(\frac{9}{70}\right)$$ $$e\left(\frac{26}{35}\right)$$ $$e\left(\frac{33}{70}\right)$$ $$e\left(\frac{1}{35}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{6}{35}\right)$$
$$\chi_{639}(10,\cdot)$$ 639.v 35 no $$1$$ $$1$$ $$e\left(\frac{32}{35}\right)$$ $$e\left(\frac{29}{35}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{17}{35}\right)$$ $$e\left(\frac{26}{35}\right)$$ $$e\left(\frac{18}{35}\right)$$ $$e\left(\frac{2}{35}\right)$$ $$e\left(\frac{33}{35}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{23}{35}\right)$$
$$\chi_{639}(11,\cdot)$$ 639.bf 210 yes $$1$$ $$1$$ $$e\left(\frac{173}{210}\right)$$ $$e\left(\frac{68}{105}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{23}{210}\right)$$ $$e\left(\frac{33}{70}\right)$$ $$e\left(\frac{2}{35}\right)$$ $$e\left(\frac{94}{105}\right)$$ $$e\left(\frac{127}{210}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{31}{105}\right)$$
$$\chi_{639}(13,\cdot)$$ 639.bd 210 yes $$-1$$ $$1$$ $$e\left(\frac{71}{105}\right)$$ $$e\left(\frac{37}{105}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{187}{210}\right)$$ $$e\left(\frac{1}{35}\right)$$ $$e\left(\frac{33}{35}\right)$$ $$e\left(\frac{127}{210}\right)$$ $$e\left(\frac{83}{210}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{74}{105}\right)$$
$$\chi_{639}(14,\cdot)$$ 639.s 30 yes $$1$$ $$1$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$
$$\chi_{639}(16,\cdot)$$ 639.bc 105 yes $$1$$ $$1$$ $$e\left(\frac{76}{105}\right)$$ $$e\left(\frac{47}{105}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{1}{105}\right)$$ $$e\left(\frac{6}{35}\right)$$ $$e\left(\frac{23}{35}\right)$$ $$e\left(\frac{31}{105}\right)$$ $$e\left(\frac{74}{105}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{94}{105}\right)$$
$$\chi_{639}(17,\cdot)$$ 639.m 10 no $$1$$ $$1$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$
$$\chi_{639}(19,\cdot)$$ 639.v 35 no $$1$$ $$1$$ $$e\left(\frac{13}{35}\right)$$ $$e\left(\frac{26}{35}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{8}{35}\right)$$ $$e\left(\frac{4}{35}\right)$$ $$e\left(\frac{27}{35}\right)$$ $$e\left(\frac{3}{35}\right)$$ $$e\left(\frac{32}{35}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{17}{35}\right)$$
$$\chi_{639}(20,\cdot)$$ 639.x 42 yes $$-1$$ $$1$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{8}{21}\right)$$
$$\chi_{639}(22,\cdot)$$ 639.bd 210 yes $$-1$$ $$1$$ $$e\left(\frac{53}{105}\right)$$ $$e\left(\frac{1}{105}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{181}{210}\right)$$ $$e\left(\frac{18}{35}\right)$$ $$e\left(\frac{34}{35}\right)$$ $$e\left(\frac{151}{210}\right)$$ $$e\left(\frac{59}{210}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{2}{105}\right)$$
$$\chi_{639}(23,\cdot)$$ 639.w 42 yes $$1$$ $$1$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{10}{21}\right)$$
$$\chi_{639}(25,\cdot)$$ 639.q 15 yes $$1$$ $$1$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{13}{15}\right)$$
$$\chi_{639}(26,\cdot)$$ 639.p 14 no $$1$$ $$1$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$-1$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$1$$ $$e\left(\frac{3}{7}\right)$$
$$\chi_{639}(28,\cdot)$$ 639.bb 70 no $$-1$$ $$1$$ $$e\left(\frac{4}{35}\right)$$ $$e\left(\frac{8}{35}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{13}{70}\right)$$ $$e\left(\frac{12}{35}\right)$$ $$e\left(\frac{11}{35}\right)$$ $$e\left(\frac{53}{70}\right)$$ $$e\left(\frac{17}{70}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{16}{35}\right)$$
$$\chi_{639}(29,\cdot)$$ 639.be 210 yes $$-1$$ $$1$$ $$e\left(\frac{209}{210}\right)$$ $$e\left(\frac{104}{105}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{67}{105}\right)$$ $$e\left(\frac{69}{70}\right)$$ $$e\left(\frac{1}{35}\right)$$ $$e\left(\frac{59}{210}\right)$$ $$e\left(\frac{23}{105}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{103}{105}\right)$$
$$\chi_{639}(31,\cdot)$$ 639.bd 210 yes $$-1$$ $$1$$ $$e\left(\frac{29}{105}\right)$$ $$e\left(\frac{58}{105}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{103}{210}\right)$$ $$e\left(\frac{29}{35}\right)$$ $$e\left(\frac{12}{35}\right)$$ $$e\left(\frac{43}{210}\right)$$ $$e\left(\frac{167}{210}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{11}{105}\right)$$
$$\chi_{639}(32,\cdot)$$ 639.x 42 yes $$-1$$ $$1$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{13}{21}\right)$$
$$\chi_{639}(34,\cdot)$$ 639.y 42 yes $$-1$$ $$1$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{11}{21}\right)$$
$$\chi_{639}(35,\cdot)$$ 639.z 70 no $$1$$ $$1$$ $$e\left(\frac{69}{70}\right)$$ $$e\left(\frac{34}{35}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{29}{70}\right)$$ $$e\left(\frac{67}{70}\right)$$ $$e\left(\frac{3}{35}\right)$$ $$e\left(\frac{12}{35}\right)$$ $$e\left(\frac{11}{70}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{33}{35}\right)$$
$$\chi_{639}(37,\cdot)$$ 639.j 7 no $$1$$ $$1$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$1$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$1$$ $$e\left(\frac{6}{7}\right)$$
$$\chi_{639}(38,\cdot)$$ 639.be 210 yes $$-1$$ $$1$$ $$e\left(\frac{11}{210}\right)$$ $$e\left(\frac{11}{105}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{103}{105}\right)$$ $$e\left(\frac{11}{70}\right)$$ $$e\left(\frac{24}{35}\right)$$ $$e\left(\frac{191}{210}\right)$$ $$e\left(\frac{62}{105}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{22}{105}\right)$$
$$\chi_{639}(40,\cdot)$$ 639.bc 105 yes $$1$$ $$1$$ $$e\left(\frac{29}{105}\right)$$ $$e\left(\frac{58}{105}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{104}{105}\right)$$ $$e\left(\frac{29}{35}\right)$$ $$e\left(\frac{12}{35}\right)$$ $$e\left(\frac{74}{105}\right)$$ $$e\left(\frac{31}{105}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{11}{105}\right)$$
$$\chi_{639}(41,\cdot)$$ 639.w 42 yes $$1$$ $$1$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{19}{21}\right)$$
$$\chi_{639}(43,\cdot)$$ 639.bc 105 yes $$1$$ $$1$$ $$e\left(\frac{82}{105}\right)$$ $$e\left(\frac{59}{105}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{37}{105}\right)$$ $$e\left(\frac{12}{35}\right)$$ $$e\left(\frac{11}{35}\right)$$ $$e\left(\frac{97}{105}\right)$$ $$e\left(\frac{8}{105}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{13}{105}\right)$$
$$\chi_{639}(44,\cdot)$$ 639.z 70 no $$1$$ $$1$$ $$e\left(\frac{13}{70}\right)$$ $$e\left(\frac{13}{35}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{43}{70}\right)$$ $$e\left(\frac{39}{70}\right)$$ $$e\left(\frac{31}{35}\right)$$ $$e\left(\frac{19}{35}\right)$$ $$e\left(\frac{67}{70}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{26}{35}\right)$$
$$\chi_{639}(46,\cdot)$$ 639.k 10 no $$-1$$ $$1$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$
$$\chi_{639}(47,\cdot)$$ 639.bf 210 yes $$1$$ $$1$$ $$e\left(\frac{197}{210}\right)$$ $$e\left(\frac{92}{105}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{167}{210}\right)$$ $$e\left(\frac{57}{70}\right)$$ $$e\left(\frac{13}{35}\right)$$ $$e\left(\frac{16}{105}\right)$$ $$e\left(\frac{73}{210}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{79}{105}\right)$$