Properties

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

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(539, base_ring=CyclotomicField(210))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([50,42]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(4,539))
 
pari: order = charorder(g,chi)
 
pari: [ 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 \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(8\) \(9\) \(10\) \(12\) \(13\)
\(\chi_{539}(4,\cdot)\) \(1\) \(1\) \(e\left(\frac{41}{105}\right)\) \(e\left(\frac{88}{105}\right)\) \(e\left(\frac{82}{105}\right)\) \(e\left(\frac{74}{105}\right)\) \(e\left(\frac{8}{35}\right)\) \(e\left(\frac{6}{35}\right)\) \(e\left(\frac{71}{105}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{2}{35}\right)\)
\(\chi_{539}(9,\cdot)\) \(1\) \(1\) \(e\left(\frac{88}{105}\right)\) \(e\left(\frac{89}{105}\right)\) \(e\left(\frac{71}{105}\right)\) \(e\left(\frac{82}{105}\right)\) \(e\left(\frac{24}{35}\right)\) \(e\left(\frac{18}{35}\right)\) \(e\left(\frac{73}{105}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{6}{35}\right)\)
\(\chi_{539}(16,\cdot)\) \(1\) \(1\) \(e\left(\frac{82}{105}\right)\) \(e\left(\frac{71}{105}\right)\) \(e\left(\frac{59}{105}\right)\) \(e\left(\frac{43}{105}\right)\) \(e\left(\frac{16}{35}\right)\) \(e\left(\frac{12}{35}\right)\) \(e\left(\frac{37}{105}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{4}{35}\right)\)
\(\chi_{539}(25,\cdot)\) \(1\) \(1\) \(e\left(\frac{74}{105}\right)\) \(e\left(\frac{82}{105}\right)\) \(e\left(\frac{43}{105}\right)\) \(e\left(\frac{26}{105}\right)\) \(e\left(\frac{17}{35}\right)\) \(e\left(\frac{4}{35}\right)\) \(e\left(\frac{59}{105}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{13}{35}\right)\)
\(\chi_{539}(37,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{105}\right)\) \(e\left(\frac{38}{105}\right)\) \(e\left(\frac{2}{105}\right)\) \(e\left(\frac{94}{105}\right)\) \(e\left(\frac{13}{35}\right)\) \(e\left(\frac{1}{35}\right)\) \(e\left(\frac{76}{105}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{12}{35}\right)\)
\(\chi_{539}(53,\cdot)\) \(1\) \(1\) \(e\left(\frac{83}{105}\right)\) \(e\left(\frac{4}{105}\right)\) \(e\left(\frac{61}{105}\right)\) \(e\left(\frac{32}{105}\right)\) \(e\left(\frac{29}{35}\right)\) \(e\left(\frac{13}{35}\right)\) \(e\left(\frac{8}{105}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{16}{35}\right)\)
\(\chi_{539}(58,\cdot)\) \(1\) \(1\) \(e\left(\frac{4}{105}\right)\) \(e\left(\frac{47}{105}\right)\) \(e\left(\frac{8}{105}\right)\) \(e\left(\frac{61}{105}\right)\) \(e\left(\frac{17}{35}\right)\) \(e\left(\frac{4}{35}\right)\) \(e\left(\frac{94}{105}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{13}{35}\right)\)
\(\chi_{539}(60,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{105}\right)\) \(e\left(\frac{16}{105}\right)\) \(e\left(\frac{34}{105}\right)\) \(e\left(\frac{23}{105}\right)\) \(e\left(\frac{11}{35}\right)\) \(e\left(\frac{17}{35}\right)\) \(e\left(\frac{32}{105}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{29}{35}\right)\)
\(\chi_{539}(81,\cdot)\) \(1\) \(1\) \(e\left(\frac{71}{105}\right)\) \(e\left(\frac{73}{105}\right)\) \(e\left(\frac{37}{105}\right)\) \(e\left(\frac{59}{105}\right)\) \(e\left(\frac{13}{35}\right)\) \(e\left(\frac{1}{35}\right)\) \(e\left(\frac{41}{105}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{12}{35}\right)\)
\(\chi_{539}(86,\cdot)\) \(1\) \(1\) \(e\left(\frac{43}{105}\right)\) \(e\left(\frac{59}{105}\right)\) \(e\left(\frac{86}{105}\right)\) \(e\left(\frac{52}{105}\right)\) \(e\left(\frac{34}{35}\right)\) \(e\left(\frac{8}{35}\right)\) \(e\left(\frac{13}{105}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{26}{35}\right)\)
\(\chi_{539}(93,\cdot)\) \(1\) \(1\) \(e\left(\frac{37}{105}\right)\) \(e\left(\frac{41}{105}\right)\) \(e\left(\frac{74}{105}\right)\) \(e\left(\frac{13}{105}\right)\) \(e\left(\frac{26}{35}\right)\) \(e\left(\frac{2}{35}\right)\) \(e\left(\frac{82}{105}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{24}{35}\right)\)
\(\chi_{539}(102,\cdot)\) \(1\) \(1\) \(e\left(\frac{104}{105}\right)\) \(e\left(\frac{67}{105}\right)\) \(e\left(\frac{103}{105}\right)\) \(e\left(\frac{11}{105}\right)\) \(e\left(\frac{22}{35}\right)\) \(e\left(\frac{34}{35}\right)\) \(e\left(\frac{29}{105}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{23}{35}\right)\)
\(\chi_{539}(114,\cdot)\) \(1\) \(1\) \(e\left(\frac{61}{105}\right)\) \(e\left(\frac{8}{105}\right)\) \(e\left(\frac{17}{105}\right)\) \(e\left(\frac{64}{105}\right)\) \(e\left(\frac{23}{35}\right)\) \(e\left(\frac{26}{35}\right)\) \(e\left(\frac{16}{105}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{32}{35}\right)\)
\(\chi_{539}(130,\cdot)\) \(1\) \(1\) \(e\left(\frac{8}{105}\right)\) \(e\left(\frac{94}{105}\right)\) \(e\left(\frac{16}{105}\right)\) \(e\left(\frac{17}{105}\right)\) \(e\left(\frac{34}{35}\right)\) \(e\left(\frac{8}{35}\right)\) \(e\left(\frac{83}{105}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{26}{35}\right)\)
\(\chi_{539}(135,\cdot)\) \(1\) \(1\) \(e\left(\frac{64}{105}\right)\) \(e\left(\frac{17}{105}\right)\) \(e\left(\frac{23}{105}\right)\) \(e\left(\frac{31}{105}\right)\) \(e\left(\frac{27}{35}\right)\) \(e\left(\frac{29}{35}\right)\) \(e\left(\frac{34}{105}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{33}{35}\right)\)
\(\chi_{539}(137,\cdot)\) \(1\) \(1\) \(e\left(\frac{47}{105}\right)\) \(e\left(\frac{1}{105}\right)\) \(e\left(\frac{94}{105}\right)\) \(e\left(\frac{8}{105}\right)\) \(e\left(\frac{16}{35}\right)\) \(e\left(\frac{12}{35}\right)\) \(e\left(\frac{2}{105}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{4}{35}\right)\)
\(\chi_{539}(158,\cdot)\) \(1\) \(1\) \(e\left(\frac{101}{105}\right)\) \(e\left(\frac{58}{105}\right)\) \(e\left(\frac{97}{105}\right)\) \(e\left(\frac{44}{105}\right)\) \(e\left(\frac{18}{35}\right)\) \(e\left(\frac{31}{35}\right)\) \(e\left(\frac{11}{105}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{22}{35}\right)\)
\(\chi_{539}(163,\cdot)\) \(1\) \(1\) \(e\left(\frac{103}{105}\right)\) \(e\left(\frac{29}{105}\right)\) \(e\left(\frac{101}{105}\right)\) \(e\left(\frac{22}{105}\right)\) \(e\left(\frac{9}{35}\right)\) \(e\left(\frac{33}{35}\right)\) \(e\left(\frac{58}{105}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{11}{35}\right)\)
\(\chi_{539}(170,\cdot)\) \(1\) \(1\) \(e\left(\frac{97}{105}\right)\) \(e\left(\frac{11}{105}\right)\) \(e\left(\frac{89}{105}\right)\) \(e\left(\frac{88}{105}\right)\) \(e\left(\frac{1}{35}\right)\) \(e\left(\frac{27}{35}\right)\) \(e\left(\frac{22}{105}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{9}{35}\right)\)
\(\chi_{539}(179,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{105}\right)\) \(e\left(\frac{52}{105}\right)\) \(e\left(\frac{58}{105}\right)\) \(e\left(\frac{101}{105}\right)\) \(e\left(\frac{27}{35}\right)\) \(e\left(\frac{29}{35}\right)\) \(e\left(\frac{104}{105}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{33}{35}\right)\)
\(\chi_{539}(191,\cdot)\) \(1\) \(1\) \(e\left(\frac{16}{105}\right)\) \(e\left(\frac{83}{105}\right)\) \(e\left(\frac{32}{105}\right)\) \(e\left(\frac{34}{105}\right)\) \(e\left(\frac{33}{35}\right)\) \(e\left(\frac{16}{35}\right)\) \(e\left(\frac{61}{105}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{17}{35}\right)\)
\(\chi_{539}(207,\cdot)\) \(1\) \(1\) \(e\left(\frac{38}{105}\right)\) \(e\left(\frac{79}{105}\right)\) \(e\left(\frac{76}{105}\right)\) \(e\left(\frac{2}{105}\right)\) \(e\left(\frac{4}{35}\right)\) \(e\left(\frac{3}{35}\right)\) \(e\left(\frac{53}{105}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{1}{35}\right)\)
\(\chi_{539}(212,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{105}\right)\) \(e\left(\frac{92}{105}\right)\) \(e\left(\frac{38}{105}\right)\) \(e\left(\frac{1}{105}\right)\) \(e\left(\frac{2}{35}\right)\) \(e\left(\frac{19}{35}\right)\) \(e\left(\frac{79}{105}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{18}{35}\right)\)
\(\chi_{539}(235,\cdot)\) \(1\) \(1\) \(e\left(\frac{26}{105}\right)\) \(e\left(\frac{43}{105}\right)\) \(e\left(\frac{52}{105}\right)\) \(e\left(\frac{29}{105}\right)\) \(e\left(\frac{23}{35}\right)\) \(e\left(\frac{26}{35}\right)\) \(e\left(\frac{86}{105}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{32}{35}\right)\)
\(\chi_{539}(240,\cdot)\) \(1\) \(1\) \(e\left(\frac{58}{105}\right)\) \(e\left(\frac{104}{105}\right)\) \(e\left(\frac{11}{105}\right)\) \(e\left(\frac{97}{105}\right)\) \(e\left(\frac{19}{35}\right)\) \(e\left(\frac{23}{35}\right)\) \(e\left(\frac{103}{105}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{31}{35}\right)\)
\(\chi_{539}(247,\cdot)\) \(1\) \(1\) \(e\left(\frac{52}{105}\right)\) \(e\left(\frac{86}{105}\right)\) \(e\left(\frac{104}{105}\right)\) \(e\left(\frac{58}{105}\right)\) \(e\left(\frac{11}{35}\right)\) \(e\left(\frac{17}{35}\right)\) \(e\left(\frac{67}{105}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{29}{35}\right)\)
\(\chi_{539}(256,\cdot)\) \(1\) \(1\) \(e\left(\frac{59}{105}\right)\) \(e\left(\frac{37}{105}\right)\) \(e\left(\frac{13}{105}\right)\) \(e\left(\frac{86}{105}\right)\) \(e\left(\frac{32}{35}\right)\) \(e\left(\frac{24}{35}\right)\) \(e\left(\frac{74}{105}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{8}{35}\right)\)
\(\chi_{539}(268,\cdot)\) \(1\) \(1\) \(e\left(\frac{76}{105}\right)\) \(e\left(\frac{53}{105}\right)\) \(e\left(\frac{47}{105}\right)\) \(e\left(\frac{4}{105}\right)\) \(e\left(\frac{8}{35}\right)\) \(e\left(\frac{6}{35}\right)\) \(e\left(\frac{1}{105}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{2}{35}\right)\)
\(\chi_{539}(284,\cdot)\) \(1\) \(1\) \(e\left(\frac{68}{105}\right)\) \(e\left(\frac{64}{105}\right)\) \(e\left(\frac{31}{105}\right)\) \(e\left(\frac{92}{105}\right)\) \(e\left(\frac{9}{35}\right)\) \(e\left(\frac{33}{35}\right)\) \(e\left(\frac{23}{105}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{11}{35}\right)\)
\(\chi_{539}(289,\cdot)\) \(1\) \(1\) \(e\left(\frac{79}{105}\right)\) \(e\left(\frac{62}{105}\right)\) \(e\left(\frac{53}{105}\right)\) \(e\left(\frac{76}{105}\right)\) \(e\left(\frac{12}{35}\right)\) \(e\left(\frac{9}{35}\right)\) \(e\left(\frac{19}{105}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{3}{35}\right)\)
\(\chi_{539}(291,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{105}\right)\) \(e\left(\frac{76}{105}\right)\) \(e\left(\frac{4}{105}\right)\) \(e\left(\frac{83}{105}\right)\) \(e\left(\frac{26}{35}\right)\) \(e\left(\frac{2}{35}\right)\) \(e\left(\frac{47}{105}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{24}{35}\right)\)