# Properties

 Label 539.bc Modulus $539$ Conductor $539$ Order $105$ Real no Primitive yes Minimal yes Parity even

# Related objects

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(539, base_ring=CyclotomicField(210))

M = H._module

chi = DirichletCharacter(H, M([50,42]))

chi.galois_orbit()

[g,chi] = znchar(Mod(4,539))

order = charorder(g,chi)

[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

## Basic properties

 Modulus: $$539$$ Conductor: $$539$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$105$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: yes sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Related number fields

 Field of values: $\Q(\zeta_{105})$ Fixed field: Number field defined by a degree 105 polynomial (not computed)

## First 31 of 48 characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$8$$ $$9$$ $$10$$ $$12$$ $$13$$
$$\chi_{539}(4,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{41}{105}\right)$$ $$e\left(\frac{88}{105}\right)$$ $$e\left(\frac{82}{105}\right)$$ $$e\left(\frac{74}{105}\right)$$ $$e\left(\frac{8}{35}\right)$$ $$e\left(\frac{6}{35}\right)$$ $$e\left(\frac{71}{105}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{2}{35}\right)$$
$$\chi_{539}(9,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{88}{105}\right)$$ $$e\left(\frac{89}{105}\right)$$ $$e\left(\frac{71}{105}\right)$$ $$e\left(\frac{82}{105}\right)$$ $$e\left(\frac{24}{35}\right)$$ $$e\left(\frac{18}{35}\right)$$ $$e\left(\frac{73}{105}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{6}{35}\right)$$
$$\chi_{539}(16,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{82}{105}\right)$$ $$e\left(\frac{71}{105}\right)$$ $$e\left(\frac{59}{105}\right)$$ $$e\left(\frac{43}{105}\right)$$ $$e\left(\frac{16}{35}\right)$$ $$e\left(\frac{12}{35}\right)$$ $$e\left(\frac{37}{105}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{4}{35}\right)$$
$$\chi_{539}(25,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{74}{105}\right)$$ $$e\left(\frac{82}{105}\right)$$ $$e\left(\frac{43}{105}\right)$$ $$e\left(\frac{26}{105}\right)$$ $$e\left(\frac{17}{35}\right)$$ $$e\left(\frac{4}{35}\right)$$ $$e\left(\frac{59}{105}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{13}{35}\right)$$
$$\chi_{539}(37,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{105}\right)$$ $$e\left(\frac{38}{105}\right)$$ $$e\left(\frac{2}{105}\right)$$ $$e\left(\frac{94}{105}\right)$$ $$e\left(\frac{13}{35}\right)$$ $$e\left(\frac{1}{35}\right)$$ $$e\left(\frac{76}{105}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{12}{35}\right)$$
$$\chi_{539}(53,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{83}{105}\right)$$ $$e\left(\frac{4}{105}\right)$$ $$e\left(\frac{61}{105}\right)$$ $$e\left(\frac{32}{105}\right)$$ $$e\left(\frac{29}{35}\right)$$ $$e\left(\frac{13}{35}\right)$$ $$e\left(\frac{8}{105}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{16}{35}\right)$$
$$\chi_{539}(58,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{4}{105}\right)$$ $$e\left(\frac{47}{105}\right)$$ $$e\left(\frac{8}{105}\right)$$ $$e\left(\frac{61}{105}\right)$$ $$e\left(\frac{17}{35}\right)$$ $$e\left(\frac{4}{35}\right)$$ $$e\left(\frac{94}{105}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{13}{35}\right)$$
$$\chi_{539}(60,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{17}{105}\right)$$ $$e\left(\frac{16}{105}\right)$$ $$e\left(\frac{34}{105}\right)$$ $$e\left(\frac{23}{105}\right)$$ $$e\left(\frac{11}{35}\right)$$ $$e\left(\frac{17}{35}\right)$$ $$e\left(\frac{32}{105}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{29}{35}\right)$$
$$\chi_{539}(81,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{71}{105}\right)$$ $$e\left(\frac{73}{105}\right)$$ $$e\left(\frac{37}{105}\right)$$ $$e\left(\frac{59}{105}\right)$$ $$e\left(\frac{13}{35}\right)$$ $$e\left(\frac{1}{35}\right)$$ $$e\left(\frac{41}{105}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{12}{35}\right)$$
$$\chi_{539}(86,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{43}{105}\right)$$ $$e\left(\frac{59}{105}\right)$$ $$e\left(\frac{86}{105}\right)$$ $$e\left(\frac{52}{105}\right)$$ $$e\left(\frac{34}{35}\right)$$ $$e\left(\frac{8}{35}\right)$$ $$e\left(\frac{13}{105}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{26}{35}\right)$$
$$\chi_{539}(93,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{37}{105}\right)$$ $$e\left(\frac{41}{105}\right)$$ $$e\left(\frac{74}{105}\right)$$ $$e\left(\frac{13}{105}\right)$$ $$e\left(\frac{26}{35}\right)$$ $$e\left(\frac{2}{35}\right)$$ $$e\left(\frac{82}{105}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{24}{35}\right)$$
$$\chi_{539}(102,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{104}{105}\right)$$ $$e\left(\frac{67}{105}\right)$$ $$e\left(\frac{103}{105}\right)$$ $$e\left(\frac{11}{105}\right)$$ $$e\left(\frac{22}{35}\right)$$ $$e\left(\frac{34}{35}\right)$$ $$e\left(\frac{29}{105}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{23}{35}\right)$$
$$\chi_{539}(114,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{61}{105}\right)$$ $$e\left(\frac{8}{105}\right)$$ $$e\left(\frac{17}{105}\right)$$ $$e\left(\frac{64}{105}\right)$$ $$e\left(\frac{23}{35}\right)$$ $$e\left(\frac{26}{35}\right)$$ $$e\left(\frac{16}{105}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{32}{35}\right)$$
$$\chi_{539}(130,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{8}{105}\right)$$ $$e\left(\frac{94}{105}\right)$$ $$e\left(\frac{16}{105}\right)$$ $$e\left(\frac{17}{105}\right)$$ $$e\left(\frac{34}{35}\right)$$ $$e\left(\frac{8}{35}\right)$$ $$e\left(\frac{83}{105}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{26}{35}\right)$$
$$\chi_{539}(135,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{64}{105}\right)$$ $$e\left(\frac{17}{105}\right)$$ $$e\left(\frac{23}{105}\right)$$ $$e\left(\frac{31}{105}\right)$$ $$e\left(\frac{27}{35}\right)$$ $$e\left(\frac{29}{35}\right)$$ $$e\left(\frac{34}{105}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{33}{35}\right)$$
$$\chi_{539}(137,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{47}{105}\right)$$ $$e\left(\frac{1}{105}\right)$$ $$e\left(\frac{94}{105}\right)$$ $$e\left(\frac{8}{105}\right)$$ $$e\left(\frac{16}{35}\right)$$ $$e\left(\frac{12}{35}\right)$$ $$e\left(\frac{2}{105}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{4}{35}\right)$$
$$\chi_{539}(158,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{101}{105}\right)$$ $$e\left(\frac{58}{105}\right)$$ $$e\left(\frac{97}{105}\right)$$ $$e\left(\frac{44}{105}\right)$$ $$e\left(\frac{18}{35}\right)$$ $$e\left(\frac{31}{35}\right)$$ $$e\left(\frac{11}{105}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{22}{35}\right)$$
$$\chi_{539}(163,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{103}{105}\right)$$ $$e\left(\frac{29}{105}\right)$$ $$e\left(\frac{101}{105}\right)$$ $$e\left(\frac{22}{105}\right)$$ $$e\left(\frac{9}{35}\right)$$ $$e\left(\frac{33}{35}\right)$$ $$e\left(\frac{58}{105}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{11}{35}\right)$$
$$\chi_{539}(170,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{97}{105}\right)$$ $$e\left(\frac{11}{105}\right)$$ $$e\left(\frac{89}{105}\right)$$ $$e\left(\frac{88}{105}\right)$$ $$e\left(\frac{1}{35}\right)$$ $$e\left(\frac{27}{35}\right)$$ $$e\left(\frac{22}{105}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{9}{35}\right)$$
$$\chi_{539}(179,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{29}{105}\right)$$ $$e\left(\frac{52}{105}\right)$$ $$e\left(\frac{58}{105}\right)$$ $$e\left(\frac{101}{105}\right)$$ $$e\left(\frac{27}{35}\right)$$ $$e\left(\frac{29}{35}\right)$$ $$e\left(\frac{104}{105}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{33}{35}\right)$$
$$\chi_{539}(191,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{16}{105}\right)$$ $$e\left(\frac{83}{105}\right)$$ $$e\left(\frac{32}{105}\right)$$ $$e\left(\frac{34}{105}\right)$$ $$e\left(\frac{33}{35}\right)$$ $$e\left(\frac{16}{35}\right)$$ $$e\left(\frac{61}{105}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{17}{35}\right)$$
$$\chi_{539}(207,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{38}{105}\right)$$ $$e\left(\frac{79}{105}\right)$$ $$e\left(\frac{76}{105}\right)$$ $$e\left(\frac{2}{105}\right)$$ $$e\left(\frac{4}{35}\right)$$ $$e\left(\frac{3}{35}\right)$$ $$e\left(\frac{53}{105}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{1}{35}\right)$$
$$\chi_{539}(212,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{19}{105}\right)$$ $$e\left(\frac{92}{105}\right)$$ $$e\left(\frac{38}{105}\right)$$ $$e\left(\frac{1}{105}\right)$$ $$e\left(\frac{2}{35}\right)$$ $$e\left(\frac{19}{35}\right)$$ $$e\left(\frac{79}{105}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{18}{35}\right)$$
$$\chi_{539}(235,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{26}{105}\right)$$ $$e\left(\frac{43}{105}\right)$$ $$e\left(\frac{52}{105}\right)$$ $$e\left(\frac{29}{105}\right)$$ $$e\left(\frac{23}{35}\right)$$ $$e\left(\frac{26}{35}\right)$$ $$e\left(\frac{86}{105}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{32}{35}\right)$$
$$\chi_{539}(240,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{58}{105}\right)$$ $$e\left(\frac{104}{105}\right)$$ $$e\left(\frac{11}{105}\right)$$ $$e\left(\frac{97}{105}\right)$$ $$e\left(\frac{19}{35}\right)$$ $$e\left(\frac{23}{35}\right)$$ $$e\left(\frac{103}{105}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{31}{35}\right)$$
$$\chi_{539}(247,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{52}{105}\right)$$ $$e\left(\frac{86}{105}\right)$$ $$e\left(\frac{104}{105}\right)$$ $$e\left(\frac{58}{105}\right)$$ $$e\left(\frac{11}{35}\right)$$ $$e\left(\frac{17}{35}\right)$$ $$e\left(\frac{67}{105}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{29}{35}\right)$$
$$\chi_{539}(256,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{59}{105}\right)$$ $$e\left(\frac{37}{105}\right)$$ $$e\left(\frac{13}{105}\right)$$ $$e\left(\frac{86}{105}\right)$$ $$e\left(\frac{32}{35}\right)$$ $$e\left(\frac{24}{35}\right)$$ $$e\left(\frac{74}{105}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{8}{35}\right)$$
$$\chi_{539}(268,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{76}{105}\right)$$ $$e\left(\frac{53}{105}\right)$$ $$e\left(\frac{47}{105}\right)$$ $$e\left(\frac{4}{105}\right)$$ $$e\left(\frac{8}{35}\right)$$ $$e\left(\frac{6}{35}\right)$$ $$e\left(\frac{1}{105}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{2}{35}\right)$$
$$\chi_{539}(284,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{68}{105}\right)$$ $$e\left(\frac{64}{105}\right)$$ $$e\left(\frac{31}{105}\right)$$ $$e\left(\frac{92}{105}\right)$$ $$e\left(\frac{9}{35}\right)$$ $$e\left(\frac{33}{35}\right)$$ $$e\left(\frac{23}{105}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{11}{35}\right)$$
$$\chi_{539}(289,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{79}{105}\right)$$ $$e\left(\frac{62}{105}\right)$$ $$e\left(\frac{53}{105}\right)$$ $$e\left(\frac{76}{105}\right)$$ $$e\left(\frac{12}{35}\right)$$ $$e\left(\frac{9}{35}\right)$$ $$e\left(\frac{19}{105}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{3}{35}\right)$$
$$\chi_{539}(291,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{2}{105}\right)$$ $$e\left(\frac{76}{105}\right)$$ $$e\left(\frac{4}{105}\right)$$ $$e\left(\frac{83}{105}\right)$$ $$e\left(\frac{26}{35}\right)$$ $$e\left(\frac{2}{35}\right)$$ $$e\left(\frac{47}{105}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{24}{35}\right)$$