# Properties

 Label 1078.bf Modulus $1078$ Conductor $539$ Order $210$ Real no Primitive no Minimal yes Parity even

# Related objects

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

sage: H = DirichletGroup(1078, base_ring=CyclotomicField(210))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([125,189]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(17,1078))

pari: order = charorder(g,chi)

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

## Basic properties

 Modulus: $$1078$$ Conductor: $$539$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$210$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from 539.bf 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 210 polynomial (not computed)

## First 31 of 48 characters in Galois orbit

Character $$-1$$ $$1$$ $$3$$ $$5$$ $$9$$ $$13$$ $$15$$ $$17$$ $$19$$ $$23$$ $$25$$ $$27$$
$$\chi_{1078}(17,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{167}{210}\right)$$ $$e\left(\frac{181}{210}\right)$$ $$e\left(\frac{62}{105}\right)$$ $$e\left(\frac{19}{35}\right)$$ $$e\left(\frac{23}{35}\right)$$ $$e\left(\frac{103}{105}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{76}{105}\right)$$ $$e\left(\frac{27}{70}\right)$$
$$\chi_{1078}(61,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{97}{210}\right)$$ $$e\left(\frac{41}{210}\right)$$ $$e\left(\frac{97}{105}\right)$$ $$e\left(\frac{19}{35}\right)$$ $$e\left(\frac{23}{35}\right)$$ $$e\left(\frac{68}{105}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{41}{105}\right)$$ $$e\left(\frac{27}{70}\right)$$
$$\chi_{1078}(73,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{101}{210}\right)$$ $$e\left(\frac{73}{210}\right)$$ $$e\left(\frac{101}{105}\right)$$ $$e\left(\frac{27}{35}\right)$$ $$e\left(\frac{29}{35}\right)$$ $$e\left(\frac{34}{105}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{73}{105}\right)$$ $$e\left(\frac{31}{70}\right)$$
$$\chi_{1078}(101,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{173}{210}\right)$$ $$e\left(\frac{19}{210}\right)$$ $$e\left(\frac{68}{105}\right)$$ $$e\left(\frac{31}{35}\right)$$ $$e\left(\frac{32}{35}\right)$$ $$e\left(\frac{52}{105}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{19}{105}\right)$$ $$e\left(\frac{33}{70}\right)$$
$$\chi_{1078}(145,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{193}{210}\right)$$ $$e\left(\frac{179}{210}\right)$$ $$e\left(\frac{88}{105}\right)$$ $$e\left(\frac{1}{35}\right)$$ $$e\left(\frac{27}{35}\right)$$ $$e\left(\frac{92}{105}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{74}{105}\right)$$ $$e\left(\frac{53}{70}\right)$$
$$\chi_{1078}(171,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{17}{210}\right)$$ $$e\left(\frac{31}{210}\right)$$ $$e\left(\frac{17}{105}\right)$$ $$e\left(\frac{34}{35}\right)$$ $$e\left(\frac{8}{35}\right)$$ $$e\left(\frac{13}{105}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{31}{105}\right)$$ $$e\left(\frac{17}{70}\right)$$
$$\chi_{1078}(173,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{169}{210}\right)$$ $$e\left(\frac{197}{210}\right)$$ $$e\left(\frac{64}{105}\right)$$ $$e\left(\frac{23}{35}\right)$$ $$e\left(\frac{26}{35}\right)$$ $$e\left(\frac{86}{105}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{92}{105}\right)$$ $$e\left(\frac{29}{70}\right)$$
$$\chi_{1078}(255,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{23}{210}\right)$$ $$e\left(\frac{79}{210}\right)$$ $$e\left(\frac{23}{105}\right)$$ $$e\left(\frac{11}{35}\right)$$ $$e\left(\frac{17}{35}\right)$$ $$e\left(\frac{67}{105}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{79}{105}\right)$$ $$e\left(\frac{23}{70}\right)$$
$$\chi_{1078}(271,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{210}\right)$$ $$e\left(\frac{113}{210}\right)$$ $$e\left(\frac{1}{105}\right)$$ $$e\left(\frac{2}{35}\right)$$ $$e\left(\frac{19}{35}\right)$$ $$e\left(\frac{44}{105}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{8}{105}\right)$$ $$e\left(\frac{1}{70}\right)$$
$$\chi_{1078}(283,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{179}{210}\right)$$ $$e\left(\frac{67}{210}\right)$$ $$e\left(\frac{74}{105}\right)$$ $$e\left(\frac{8}{35}\right)$$ $$e\left(\frac{6}{35}\right)$$ $$e\left(\frac{1}{105}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{67}{105}\right)$$ $$e\left(\frac{39}{70}\right)$$
$$\chi_{1078}(299,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{103}{210}\right)$$ $$e\left(\frac{89}{210}\right)$$ $$e\left(\frac{103}{105}\right)$$ $$e\left(\frac{31}{35}\right)$$ $$e\left(\frac{32}{35}\right)$$ $$e\left(\frac{17}{105}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{89}{105}\right)$$ $$e\left(\frac{33}{70}\right)$$
$$\chi_{1078}(327,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{79}{210}\right)$$ $$e\left(\frac{107}{210}\right)$$ $$e\left(\frac{79}{105}\right)$$ $$e\left(\frac{18}{35}\right)$$ $$e\left(\frac{31}{35}\right)$$ $$e\left(\frac{11}{105}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{2}{105}\right)$$ $$e\left(\frac{9}{70}\right)$$
$$\chi_{1078}(369,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{127}{210}\right)$$ $$e\left(\frac{71}{210}\right)$$ $$e\left(\frac{22}{105}\right)$$ $$e\left(\frac{9}{35}\right)$$ $$e\left(\frac{33}{35}\right)$$ $$e\left(\frac{23}{105}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{71}{105}\right)$$ $$e\left(\frac{57}{70}\right)$$
$$\chi_{1078}(381,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{11}{210}\right)$$ $$e\left(\frac{193}{210}\right)$$ $$e\left(\frac{11}{105}\right)$$ $$e\left(\frac{22}{35}\right)$$ $$e\left(\frac{34}{35}\right)$$ $$e\left(\frac{64}{105}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{88}{105}\right)$$ $$e\left(\frac{11}{70}\right)$$
$$\chi_{1078}(409,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{83}{210}\right)$$ $$e\left(\frac{139}{210}\right)$$ $$e\left(\frac{83}{105}\right)$$ $$e\left(\frac{26}{35}\right)$$ $$e\left(\frac{2}{35}\right)$$ $$e\left(\frac{82}{105}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{34}{105}\right)$$ $$e\left(\frac{13}{70}\right)$$
$$\chi_{1078}(425,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{121}{210}\right)$$ $$e\left(\frac{23}{210}\right)$$ $$e\left(\frac{16}{105}\right)$$ $$e\left(\frac{32}{35}\right)$$ $$e\left(\frac{24}{35}\right)$$ $$e\left(\frac{74}{105}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{23}{105}\right)$$ $$e\left(\frac{51}{70}\right)$$
$$\chi_{1078}(437,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{29}{210}\right)$$ $$e\left(\frac{127}{210}\right)$$ $$e\left(\frac{29}{105}\right)$$ $$e\left(\frac{23}{35}\right)$$ $$e\left(\frac{26}{35}\right)$$ $$e\left(\frac{16}{105}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{22}{105}\right)$$ $$e\left(\frac{29}{70}\right)$$
$$\chi_{1078}(453,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{13}{210}\right)$$ $$e\left(\frac{209}{210}\right)$$ $$e\left(\frac{13}{105}\right)$$ $$e\left(\frac{26}{35}\right)$$ $$e\left(\frac{2}{35}\right)$$ $$e\left(\frac{47}{105}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{104}{105}\right)$$ $$e\left(\frac{13}{70}\right)$$
$$\chi_{1078}(479,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{137}{210}\right)$$ $$e\left(\frac{151}{210}\right)$$ $$e\left(\frac{32}{105}\right)$$ $$e\left(\frac{29}{35}\right)$$ $$e\left(\frac{13}{35}\right)$$ $$e\left(\frac{43}{105}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{46}{105}\right)$$ $$e\left(\frac{67}{70}\right)$$
$$\chi_{1078}(481,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{199}{210}\right)$$ $$e\left(\frac{17}{210}\right)$$ $$e\left(\frac{94}{105}\right)$$ $$e\left(\frac{13}{35}\right)$$ $$e\left(\frac{1}{35}\right)$$ $$e\left(\frac{41}{105}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{17}{105}\right)$$ $$e\left(\frac{59}{70}\right)$$
$$\chi_{1078}(523,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{37}{210}\right)$$ $$e\left(\frac{191}{210}\right)$$ $$e\left(\frac{37}{105}\right)$$ $$e\left(\frac{4}{35}\right)$$ $$e\left(\frac{3}{35}\right)$$ $$e\left(\frac{53}{105}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{86}{105}\right)$$ $$e\left(\frac{37}{70}\right)$$
$$\chi_{1078}(535,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{71}{210}\right)$$ $$e\left(\frac{43}{210}\right)$$ $$e\left(\frac{71}{105}\right)$$ $$e\left(\frac{2}{35}\right)$$ $$e\left(\frac{19}{35}\right)$$ $$e\left(\frac{79}{105}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{43}{105}\right)$$ $$e\left(\frac{1}{70}\right)$$
$$\chi_{1078}(563,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{143}{210}\right)$$ $$e\left(\frac{199}{210}\right)$$ $$e\left(\frac{38}{105}\right)$$ $$e\left(\frac{6}{35}\right)$$ $$e\left(\frac{22}{35}\right)$$ $$e\left(\frac{97}{105}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{94}{105}\right)$$ $$e\left(\frac{3}{70}\right)$$
$$\chi_{1078}(579,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{31}{210}\right)$$ $$e\left(\frac{143}{210}\right)$$ $$e\left(\frac{31}{105}\right)$$ $$e\left(\frac{27}{35}\right)$$ $$e\left(\frac{29}{35}\right)$$ $$e\left(\frac{104}{105}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{38}{105}\right)$$ $$e\left(\frac{31}{70}\right)$$
$$\chi_{1078}(591,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{89}{210}\right)$$ $$e\left(\frac{187}{210}\right)$$ $$e\left(\frac{89}{105}\right)$$ $$e\left(\frac{3}{35}\right)$$ $$e\left(\frac{11}{35}\right)$$ $$e\left(\frac{31}{105}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{82}{105}\right)$$ $$e\left(\frac{19}{70}\right)$$
$$\chi_{1078}(633,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{197}{210}\right)$$ $$e\left(\frac{1}{210}\right)$$ $$e\left(\frac{92}{105}\right)$$ $$e\left(\frac{9}{35}\right)$$ $$e\left(\frac{33}{35}\right)$$ $$e\left(\frac{58}{105}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{1}{105}\right)$$ $$e\left(\frac{57}{70}\right)$$
$$\chi_{1078}(635,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{109}{210}\right)$$ $$e\left(\frac{137}{210}\right)$$ $$e\left(\frac{4}{105}\right)$$ $$e\left(\frac{8}{35}\right)$$ $$e\left(\frac{6}{35}\right)$$ $$e\left(\frac{71}{105}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{32}{105}\right)$$ $$e\left(\frac{39}{70}\right)$$
$$\chi_{1078}(677,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{157}{210}\right)$$ $$e\left(\frac{101}{210}\right)$$ $$e\left(\frac{52}{105}\right)$$ $$e\left(\frac{34}{35}\right)$$ $$e\left(\frac{8}{35}\right)$$ $$e\left(\frac{83}{105}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{101}{105}\right)$$ $$e\left(\frac{17}{70}\right)$$
$$\chi_{1078}(689,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{131}{210}\right)$$ $$e\left(\frac{103}{210}\right)$$ $$e\left(\frac{26}{105}\right)$$ $$e\left(\frac{17}{35}\right)$$ $$e\left(\frac{4}{35}\right)$$ $$e\left(\frac{94}{105}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{103}{105}\right)$$ $$e\left(\frac{61}{70}\right)$$
$$\chi_{1078}(733,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{151}{210}\right)$$ $$e\left(\frac{53}{210}\right)$$ $$e\left(\frac{46}{105}\right)$$ $$e\left(\frac{22}{35}\right)$$ $$e\left(\frac{34}{35}\right)$$ $$e\left(\frac{29}{105}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{53}{105}\right)$$ $$e\left(\frac{11}{70}\right)$$
$$\chi_{1078}(745,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{149}{210}\right)$$ $$e\left(\frac{37}{210}\right)$$ $$e\left(\frac{44}{105}\right)$$ $$e\left(\frac{18}{35}\right)$$ $$e\left(\frac{31}{35}\right)$$ $$e\left(\frac{46}{105}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{37}{105}\right)$$ $$e\left(\frac{9}{70}\right)$$