# Properties

 Label 1089.bc Modulus $1089$ Conductor $1089$ Order $165$ Real no Primitive yes Minimal yes Parity even

# Related objects

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

sage: H = DirichletGroup(1089, base_ring=CyclotomicField(330))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([110,6]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(4,1089))

pari: order = charorder(g,chi)

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

## Basic properties

 Modulus: $$1089$$ Conductor: $$1089$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$165$$ 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: $\Q(\zeta_{165})$ Fixed field: Number field defined by a degree 165 polynomial (not computed)

## First 31 of 80 characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$13$$ $$14$$ $$16$$ $$17$$
$$\chi_{1089}(4,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{58}{165}\right)$$ $$e\left(\frac{116}{165}\right)$$ $$e\left(\frac{2}{165}\right)$$ $$e\left(\frac{76}{165}\right)$$ $$e\left(\frac{3}{55}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{83}{165}\right)$$ $$e\left(\frac{134}{165}\right)$$ $$e\left(\frac{67}{165}\right)$$ $$e\left(\frac{49}{55}\right)$$
$$\chi_{1089}(16,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{116}{165}\right)$$ $$e\left(\frac{67}{165}\right)$$ $$e\left(\frac{4}{165}\right)$$ $$e\left(\frac{152}{165}\right)$$ $$e\left(\frac{6}{55}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{1}{165}\right)$$ $$e\left(\frac{103}{165}\right)$$ $$e\left(\frac{134}{165}\right)$$ $$e\left(\frac{43}{55}\right)$$
$$\chi_{1089}(25,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{2}{165}\right)$$ $$e\left(\frac{4}{165}\right)$$ $$e\left(\frac{148}{165}\right)$$ $$e\left(\frac{14}{165}\right)$$ $$e\left(\frac{2}{55}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{37}{165}\right)$$ $$e\left(\frac{16}{165}\right)$$ $$e\left(\frac{8}{165}\right)$$ $$e\left(\frac{51}{55}\right)$$
$$\chi_{1089}(31,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{19}{165}\right)$$ $$e\left(\frac{38}{165}\right)$$ $$e\left(\frac{86}{165}\right)$$ $$e\left(\frac{133}{165}\right)$$ $$e\left(\frac{19}{55}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{104}{165}\right)$$ $$e\left(\frac{152}{165}\right)$$ $$e\left(\frac{76}{165}\right)$$ $$e\left(\frac{17}{55}\right)$$
$$\chi_{1089}(49,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{76}{165}\right)$$ $$e\left(\frac{152}{165}\right)$$ $$e\left(\frac{14}{165}\right)$$ $$e\left(\frac{37}{165}\right)$$ $$e\left(\frac{21}{55}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{86}{165}\right)$$ $$e\left(\frac{113}{165}\right)$$ $$e\left(\frac{139}{165}\right)$$ $$e\left(\frac{13}{55}\right)$$
$$\chi_{1089}(58,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{82}{165}\right)$$ $$e\left(\frac{164}{165}\right)$$ $$e\left(\frac{128}{165}\right)$$ $$e\left(\frac{79}{165}\right)$$ $$e\left(\frac{27}{55}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{32}{165}\right)$$ $$e\left(\frac{161}{165}\right)$$ $$e\left(\frac{163}{165}\right)$$ $$e\left(\frac{1}{55}\right)$$
$$\chi_{1089}(70,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{68}{165}\right)$$ $$e\left(\frac{136}{165}\right)$$ $$e\left(\frac{82}{165}\right)$$ $$e\left(\frac{146}{165}\right)$$ $$e\left(\frac{13}{55}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{103}{165}\right)$$ $$e\left(\frac{49}{165}\right)$$ $$e\left(\frac{107}{165}\right)$$ $$e\left(\frac{29}{55}\right)$$
$$\chi_{1089}(97,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{164}{165}\right)$$ $$e\left(\frac{163}{165}\right)$$ $$e\left(\frac{91}{165}\right)$$ $$e\left(\frac{158}{165}\right)$$ $$e\left(\frac{54}{55}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{64}{165}\right)$$ $$e\left(\frac{157}{165}\right)$$ $$e\left(\frac{161}{165}\right)$$ $$e\left(\frac{2}{55}\right)$$
$$\chi_{1089}(103,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{73}{165}\right)$$ $$e\left(\frac{146}{165}\right)$$ $$e\left(\frac{122}{165}\right)$$ $$e\left(\frac{16}{165}\right)$$ $$e\left(\frac{18}{55}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{113}{165}\right)$$ $$e\left(\frac{89}{165}\right)$$ $$e\left(\frac{127}{165}\right)$$ $$e\left(\frac{19}{55}\right)$$
$$\chi_{1089}(115,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{161}{165}\right)$$ $$e\left(\frac{157}{165}\right)$$ $$e\left(\frac{34}{165}\right)$$ $$e\left(\frac{137}{165}\right)$$ $$e\left(\frac{51}{55}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{91}{165}\right)$$ $$e\left(\frac{133}{165}\right)$$ $$e\left(\frac{149}{165}\right)$$ $$e\left(\frac{8}{55}\right)$$
$$\chi_{1089}(157,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{157}{165}\right)$$ $$e\left(\frac{149}{165}\right)$$ $$e\left(\frac{68}{165}\right)$$ $$e\left(\frac{109}{165}\right)$$ $$e\left(\frac{47}{55}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{17}{165}\right)$$ $$e\left(\frac{101}{165}\right)$$ $$e\left(\frac{133}{165}\right)$$ $$e\left(\frac{16}{55}\right)$$
$$\chi_{1089}(169,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{83}{165}\right)$$ $$e\left(\frac{1}{165}\right)$$ $$e\left(\frac{37}{165}\right)$$ $$e\left(\frac{86}{165}\right)$$ $$e\left(\frac{28}{55}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{133}{165}\right)$$ $$e\left(\frac{4}{165}\right)$$ $$e\left(\frac{2}{165}\right)$$ $$e\left(\frac{54}{55}\right)$$
$$\chi_{1089}(196,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{134}{165}\right)$$ $$e\left(\frac{103}{165}\right)$$ $$e\left(\frac{16}{165}\right)$$ $$e\left(\frac{113}{165}\right)$$ $$e\left(\frac{24}{55}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{4}{165}\right)$$ $$e\left(\frac{82}{165}\right)$$ $$e\left(\frac{41}{165}\right)$$ $$e\left(\frac{7}{55}\right)$$
$$\chi_{1089}(214,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{41}{165}\right)$$ $$e\left(\frac{82}{165}\right)$$ $$e\left(\frac{64}{165}\right)$$ $$e\left(\frac{122}{165}\right)$$ $$e\left(\frac{41}{55}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{16}{165}\right)$$ $$e\left(\frac{163}{165}\right)$$ $$e\left(\frac{164}{165}\right)$$ $$e\left(\frac{28}{55}\right)$$
$$\chi_{1089}(223,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{152}{165}\right)$$ $$e\left(\frac{139}{165}\right)$$ $$e\left(\frac{28}{165}\right)$$ $$e\left(\frac{74}{165}\right)$$ $$e\left(\frac{42}{55}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{7}{165}\right)$$ $$e\left(\frac{61}{165}\right)$$ $$e\left(\frac{113}{165}\right)$$ $$e\left(\frac{26}{55}\right)$$
$$\chi_{1089}(229,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{124}{165}\right)$$ $$e\left(\frac{83}{165}\right)$$ $$e\left(\frac{101}{165}\right)$$ $$e\left(\frac{43}{165}\right)$$ $$e\left(\frac{14}{55}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{149}{165}\right)$$ $$e\left(\frac{2}{165}\right)$$ $$e\left(\frac{1}{165}\right)$$ $$e\left(\frac{27}{55}\right)$$
$$\chi_{1089}(247,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{165}\right)$$ $$e\left(\frac{2}{165}\right)$$ $$e\left(\frac{74}{165}\right)$$ $$e\left(\frac{7}{165}\right)$$ $$e\left(\frac{1}{55}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{101}{165}\right)$$ $$e\left(\frac{8}{165}\right)$$ $$e\left(\frac{4}{165}\right)$$ $$e\left(\frac{53}{55}\right)$$
$$\chi_{1089}(256,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{67}{165}\right)$$ $$e\left(\frac{134}{165}\right)$$ $$e\left(\frac{8}{165}\right)$$ $$e\left(\frac{139}{165}\right)$$ $$e\left(\frac{12}{55}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{2}{165}\right)$$ $$e\left(\frac{41}{165}\right)$$ $$e\left(\frac{103}{165}\right)$$ $$e\left(\frac{31}{55}\right)$$
$$\chi_{1089}(268,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{98}{165}\right)$$ $$e\left(\frac{31}{165}\right)$$ $$e\left(\frac{157}{165}\right)$$ $$e\left(\frac{26}{165}\right)$$ $$e\left(\frac{43}{55}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{163}{165}\right)$$ $$e\left(\frac{124}{165}\right)$$ $$e\left(\frac{62}{165}\right)$$ $$e\left(\frac{24}{55}\right)$$
$$\chi_{1089}(295,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{104}{165}\right)$$ $$e\left(\frac{43}{165}\right)$$ $$e\left(\frac{106}{165}\right)$$ $$e\left(\frac{68}{165}\right)$$ $$e\left(\frac{49}{55}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{109}{165}\right)$$ $$e\left(\frac{7}{165}\right)$$ $$e\left(\frac{86}{165}\right)$$ $$e\left(\frac{12}{55}\right)$$
$$\chi_{1089}(301,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{103}{165}\right)$$ $$e\left(\frac{41}{165}\right)$$ $$e\left(\frac{32}{165}\right)$$ $$e\left(\frac{61}{165}\right)$$ $$e\left(\frac{48}{55}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{8}{165}\right)$$ $$e\left(\frac{164}{165}\right)$$ $$e\left(\frac{82}{165}\right)$$ $$e\left(\frac{14}{55}\right)$$
$$\chi_{1089}(313,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{86}{165}\right)$$ $$e\left(\frac{7}{165}\right)$$ $$e\left(\frac{94}{165}\right)$$ $$e\left(\frac{107}{165}\right)$$ $$e\left(\frac{31}{55}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{106}{165}\right)$$ $$e\left(\frac{28}{165}\right)$$ $$e\left(\frac{14}{165}\right)$$ $$e\left(\frac{48}{55}\right)$$
$$\chi_{1089}(322,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{62}{165}\right)$$ $$e\left(\frac{124}{165}\right)$$ $$e\left(\frac{133}{165}\right)$$ $$e\left(\frac{104}{165}\right)$$ $$e\left(\frac{7}{55}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{157}{165}\right)$$ $$e\left(\frac{1}{165}\right)$$ $$e\left(\frac{83}{165}\right)$$ $$e\left(\frac{41}{55}\right)$$
$$\chi_{1089}(328,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{94}{165}\right)$$ $$e\left(\frac{23}{165}\right)$$ $$e\left(\frac{26}{165}\right)$$ $$e\left(\frac{163}{165}\right)$$ $$e\left(\frac{39}{55}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{89}{165}\right)$$ $$e\left(\frac{92}{165}\right)$$ $$e\left(\frac{46}{165}\right)$$ $$e\left(\frac{32}{55}\right)$$
$$\chi_{1089}(346,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{46}{165}\right)$$ $$e\left(\frac{92}{165}\right)$$ $$e\left(\frac{104}{165}\right)$$ $$e\left(\frac{157}{165}\right)$$ $$e\left(\frac{46}{55}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{26}{165}\right)$$ $$e\left(\frac{38}{165}\right)$$ $$e\left(\frac{19}{165}\right)$$ $$e\left(\frac{18}{55}\right)$$
$$\chi_{1089}(355,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{142}{165}\right)$$ $$e\left(\frac{119}{165}\right)$$ $$e\left(\frac{113}{165}\right)$$ $$e\left(\frac{4}{165}\right)$$ $$e\left(\frac{32}{55}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{152}{165}\right)$$ $$e\left(\frac{146}{165}\right)$$ $$e\left(\frac{73}{165}\right)$$ $$e\left(\frac{46}{55}\right)$$
$$\chi_{1089}(367,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{113}{165}\right)$$ $$e\left(\frac{61}{165}\right)$$ $$e\left(\frac{112}{165}\right)$$ $$e\left(\frac{131}{165}\right)$$ $$e\left(\frac{3}{55}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{28}{165}\right)$$ $$e\left(\frac{79}{165}\right)$$ $$e\left(\frac{122}{165}\right)$$ $$e\left(\frac{49}{55}\right)$$
$$\chi_{1089}(394,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{74}{165}\right)$$ $$e\left(\frac{148}{165}\right)$$ $$e\left(\frac{31}{165}\right)$$ $$e\left(\frac{23}{165}\right)$$ $$e\left(\frac{19}{55}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{49}{165}\right)$$ $$e\left(\frac{97}{165}\right)$$ $$e\left(\frac{131}{165}\right)$$ $$e\left(\frac{17}{55}\right)$$
$$\chi_{1089}(400,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{118}{165}\right)$$ $$e\left(\frac{71}{165}\right)$$ $$e\left(\frac{152}{165}\right)$$ $$e\left(\frac{1}{165}\right)$$ $$e\left(\frac{8}{55}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{38}{165}\right)$$ $$e\left(\frac{119}{165}\right)$$ $$e\left(\frac{142}{165}\right)$$ $$e\left(\frac{39}{55}\right)$$
$$\chi_{1089}(412,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{131}{165}\right)$$ $$e\left(\frac{97}{165}\right)$$ $$e\left(\frac{124}{165}\right)$$ $$e\left(\frac{92}{165}\right)$$ $$e\left(\frac{21}{55}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{31}{165}\right)$$ $$e\left(\frac{58}{165}\right)$$ $$e\left(\frac{29}{165}\right)$$ $$e\left(\frac{13}{55}\right)$$
$$\chi_{1089}(421,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{137}{165}\right)$$ $$e\left(\frac{109}{165}\right)$$ $$e\left(\frac{73}{165}\right)$$ $$e\left(\frac{134}{165}\right)$$ $$e\left(\frac{27}{55}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{142}{165}\right)$$ $$e\left(\frac{106}{165}\right)$$ $$e\left(\frac{53}{165}\right)$$ $$e\left(\frac{1}{55}\right)$$