Properties

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

Related objects

Learn more

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