# Properties

 Modulus $539$ Structure $$C_{2}\times C_{210}$$ Order $420$

# Learn more

Show commands: PariGP / SageMath

sage: H = DirichletGroup(539)

pari: g = idealstar(,539,2)

## 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 = $\chi_{539}(199,\cdot)$, $\chi_{539}(442,\cdot)$

## 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 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$8$$ $$9$$ $$10$$ $$12$$ $$13$$
$$\chi_{539}(1,\cdot)$$ 539.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{539}(2,\cdot)$$ 539.be 210 yes $$-1$$ $$1$$ $$e\left(\frac{41}{210}\right)$$ $$e\left(\frac{44}{105}\right)$$ $$e\left(\frac{41}{105}\right)$$ $$e\left(\frac{37}{105}\right)$$ $$e\left(\frac{43}{70}\right)$$ $$e\left(\frac{41}{70}\right)$$ $$e\left(\frac{88}{105}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{37}{70}\right)$$
$$\chi_{539}(3,\cdot)$$ 539.bd 210 yes $$-1$$ $$1$$ $$e\left(\frac{44}{105}\right)$$ $$e\left(\frac{89}{210}\right)$$ $$e\left(\frac{88}{105}\right)$$ $$e\left(\frac{187}{210}\right)$$ $$e\left(\frac{59}{70}\right)$$ $$e\left(\frac{9}{35}\right)$$ $$e\left(\frac{89}{105}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{41}{70}\right)$$
$$\chi_{539}(4,\cdot)$$ 539.bc 105 yes $$1$$ $$1$$ $$e\left(\frac{41}{105}\right)$$ $$e\left(\frac{88}{105}\right)$$ $$e\left(\frac{82}{105}\right)$$ $$e\left(\frac{74}{105}\right)$$ $$e\left(\frac{8}{35}\right)$$ $$e\left(\frac{6}{35}\right)$$ $$e\left(\frac{71}{105}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{2}{35}\right)$$
$$\chi_{539}(5,\cdot)$$ 539.bd 210 yes $$-1$$ $$1$$ $$e\left(\frac{37}{105}\right)$$ $$e\left(\frac{187}{210}\right)$$ $$e\left(\frac{74}{105}\right)$$ $$e\left(\frac{131}{210}\right)$$ $$e\left(\frac{17}{70}\right)$$ $$e\left(\frac{2}{35}\right)$$ $$e\left(\frac{82}{105}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{13}{70}\right)$$
$$\chi_{539}(6,\cdot)$$ 539.z 70 yes $$1$$ $$1$$ $$e\left(\frac{43}{70}\right)$$ $$e\left(\frac{59}{70}\right)$$ $$e\left(\frac{8}{35}\right)$$ $$e\left(\frac{17}{70}\right)$$ $$e\left(\frac{16}{35}\right)$$ $$e\left(\frac{59}{70}\right)$$ $$e\left(\frac{24}{35}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{4}{35}\right)$$
$$\chi_{539}(8,\cdot)$$ 539.ba 70 yes $$-1$$ $$1$$ $$e\left(\frac{41}{70}\right)$$ $$e\left(\frac{9}{35}\right)$$ $$e\left(\frac{6}{35}\right)$$ $$e\left(\frac{2}{35}\right)$$ $$e\left(\frac{59}{70}\right)$$ $$e\left(\frac{53}{70}\right)$$ $$e\left(\frac{18}{35}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{41}{70}\right)$$
$$\chi_{539}(9,\cdot)$$ 539.bc 105 yes $$1$$ $$1$$ $$e\left(\frac{88}{105}\right)$$ $$e\left(\frac{89}{105}\right)$$ $$e\left(\frac{71}{105}\right)$$ $$e\left(\frac{82}{105}\right)$$ $$e\left(\frac{24}{35}\right)$$ $$e\left(\frac{18}{35}\right)$$ $$e\left(\frac{73}{105}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{6}{35}\right)$$
$$\chi_{539}(10,\cdot)$$ 539.w 42 yes $$1$$ $$1$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{5}{7}\right)$$
$$\chi_{539}(12,\cdot)$$ 539.y 42 no $$-1$$ $$1$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{9}{14}\right)$$
$$\chi_{539}(13,\cdot)$$ 539.z 70 yes $$1$$ $$1$$ $$e\left(\frac{37}{70}\right)$$ $$e\left(\frac{41}{70}\right)$$ $$e\left(\frac{2}{35}\right)$$ $$e\left(\frac{13}{70}\right)$$ $$e\left(\frac{4}{35}\right)$$ $$e\left(\frac{41}{70}\right)$$ $$e\left(\frac{6}{35}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{1}{35}\right)$$
$$\chi_{539}(15,\cdot)$$ 539.v 35 yes $$1$$ $$1$$ $$e\left(\frac{27}{35}\right)$$ $$e\left(\frac{11}{35}\right)$$ $$e\left(\frac{19}{35}\right)$$ $$e\left(\frac{18}{35}\right)$$ $$e\left(\frac{3}{35}\right)$$ $$e\left(\frac{11}{35}\right)$$ $$e\left(\frac{22}{35}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{27}{35}\right)$$
$$\chi_{539}(16,\cdot)$$ 539.bc 105 yes $$1$$ $$1$$ $$e\left(\frac{82}{105}\right)$$ $$e\left(\frac{71}{105}\right)$$ $$e\left(\frac{59}{105}\right)$$ $$e\left(\frac{43}{105}\right)$$ $$e\left(\frac{16}{35}\right)$$ $$e\left(\frac{12}{35}\right)$$ $$e\left(\frac{37}{105}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{4}{35}\right)$$
$$\chi_{539}(17,\cdot)$$ 539.bf 210 yes $$1$$ $$1$$ $$e\left(\frac{79}{210}\right)$$ $$e\left(\frac{167}{210}\right)$$ $$e\left(\frac{79}{105}\right)$$ $$e\left(\frac{181}{210}\right)$$ $$e\left(\frac{6}{35}\right)$$ $$e\left(\frac{9}{70}\right)$$ $$e\left(\frac{62}{105}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{19}{35}\right)$$
$$\chi_{539}(18,\cdot)$$ 539.t 30 no $$-1$$ $$1$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{10}\right)$$
$$\chi_{539}(19,\cdot)$$ 539.s 30 no $$1$$ $$1$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{4}{5}\right)$$
$$\chi_{539}(20,\cdot)$$ 539.bb 70 yes $$-1$$ $$1$$ $$e\left(\frac{26}{35}\right)$$ $$e\left(\frac{51}{70}\right)$$ $$e\left(\frac{17}{35}\right)$$ $$e\left(\frac{23}{70}\right)$$ $$e\left(\frac{33}{70}\right)$$ $$e\left(\frac{8}{35}\right)$$ $$e\left(\frac{16}{35}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{17}{70}\right)$$
$$\chi_{539}(23,\cdot)$$ 539.r 21 no $$1$$ $$1$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{6}{7}\right)$$
$$\chi_{539}(24,\cdot)$$ 539.bf 210 yes $$1$$ $$1$$ $$e\left(\frac{1}{210}\right)$$ $$e\left(\frac{143}{210}\right)$$ $$e\left(\frac{1}{105}\right)$$ $$e\left(\frac{199}{210}\right)$$ $$e\left(\frac{24}{35}\right)$$ $$e\left(\frac{1}{70}\right)$$ $$e\left(\frac{38}{105}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{6}{35}\right)$$
$$\chi_{539}(25,\cdot)$$ 539.bc 105 yes $$1$$ $$1$$ $$e\left(\frac{74}{105}\right)$$ $$e\left(\frac{82}{105}\right)$$ $$e\left(\frac{43}{105}\right)$$ $$e\left(\frac{26}{105}\right)$$ $$e\left(\frac{17}{35}\right)$$ $$e\left(\frac{4}{35}\right)$$ $$e\left(\frac{59}{105}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{13}{35}\right)$$
$$\chi_{539}(26,\cdot)$$ 539.bd 210 yes $$-1$$ $$1$$ $$e\left(\frac{76}{105}\right)$$ $$e\left(\frac{1}{210}\right)$$ $$e\left(\frac{47}{105}\right)$$ $$e\left(\frac{113}{210}\right)$$ $$e\left(\frac{51}{70}\right)$$ $$e\left(\frac{6}{35}\right)$$ $$e\left(\frac{1}{105}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{39}{70}\right)$$
$$\chi_{539}(27,\cdot)$$ 539.bb 70 yes $$-1$$ $$1$$ $$e\left(\frac{9}{35}\right)$$ $$e\left(\frac{19}{70}\right)$$ $$e\left(\frac{18}{35}\right)$$ $$e\left(\frac{47}{70}\right)$$ $$e\left(\frac{37}{70}\right)$$ $$e\left(\frac{27}{35}\right)$$ $$e\left(\frac{19}{35}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{53}{70}\right)$$
$$\chi_{539}(29,\cdot)$$ 539.ba 70 yes $$-1$$ $$1$$ $$e\left(\frac{59}{70}\right)$$ $$e\left(\frac{1}{35}\right)$$ $$e\left(\frac{24}{35}\right)$$ $$e\left(\frac{8}{35}\right)$$ $$e\left(\frac{61}{70}\right)$$ $$e\left(\frac{37}{70}\right)$$ $$e\left(\frac{2}{35}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{59}{70}\right)$$
$$\chi_{539}(30,\cdot)$$ 539.t 30 no $$-1$$ $$1$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{539}(31,\cdot)$$ 539.u 30 no $$-1$$ $$1$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{10}\right)$$
$$\chi_{539}(32,\cdot)$$ 539.x 42 yes $$-1$$ $$1$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{9}{14}\right)$$
$$\chi_{539}(34,\cdot)$$ 539.n 14 no $$-1$$ $$1$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{1}{14}\right)$$
$$\chi_{539}(36,\cdot)$$ 539.v 35 yes $$1$$ $$1$$ $$e\left(\frac{8}{35}\right)$$ $$e\left(\frac{24}{35}\right)$$ $$e\left(\frac{16}{35}\right)$$ $$e\left(\frac{17}{35}\right)$$ $$e\left(\frac{32}{35}\right)$$ $$e\left(\frac{24}{35}\right)$$ $$e\left(\frac{13}{35}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{8}{35}\right)$$
$$\chi_{539}(37,\cdot)$$ 539.bc 105 yes $$1$$ $$1$$ $$e\left(\frac{1}{105}\right)$$ $$e\left(\frac{38}{105}\right)$$ $$e\left(\frac{2}{105}\right)$$ $$e\left(\frac{94}{105}\right)$$ $$e\left(\frac{13}{35}\right)$$ $$e\left(\frac{1}{35}\right)$$ $$e\left(\frac{76}{105}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{12}{35}\right)$$
$$\chi_{539}(38,\cdot)$$ 539.bd 210 yes $$-1$$ $$1$$ $$e\left(\frac{17}{105}\right)$$ $$e\left(\frac{137}{210}\right)$$ $$e\left(\frac{34}{105}\right)$$ $$e\left(\frac{151}{210}\right)$$ $$e\left(\frac{57}{70}\right)$$ $$e\left(\frac{17}{35}\right)$$ $$e\left(\frac{32}{105}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{23}{70}\right)$$
$$\chi_{539}(39,\cdot)$$ 539.be 210 yes $$-1$$ $$1$$ $$e\left(\frac{199}{210}\right)$$ $$e\left(\frac{1}{105}\right)$$ $$e\left(\frac{94}{105}\right)$$ $$e\left(\frac{8}{105}\right)$$ $$e\left(\frac{67}{70}\right)$$ $$e\left(\frac{59}{70}\right)$$ $$e\left(\frac{2}{105}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{43}{70}\right)$$
$$\chi_{539}(40,\cdot)$$ 539.bf 210 yes $$1$$ $$1$$ $$e\left(\frac{197}{210}\right)$$ $$e\left(\frac{31}{210}\right)$$ $$e\left(\frac{92}{105}\right)$$ $$e\left(\frac{143}{210}\right)$$ $$e\left(\frac{3}{35}\right)$$ $$e\left(\frac{57}{70}\right)$$ $$e\left(\frac{31}{105}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{27}{35}\right)$$
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