Properties

Label 6003.dn
Modulus $6003$
Conductor $6003$
Order $462$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6003, base_ring=CyclotomicField(462))
 
M = H._module
 
chi = DirichletCharacter(H, M([308,189,363]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(34,6003))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(6003\)
Conductor: \(6003\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(462\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Related number fields

Field of values: $\Q(\zeta_{231})$
Fixed field: Number field defined by a degree 462 polynomial (not computed)

First 31 of 120 characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(4\) \(5\) \(7\) \(8\) \(10\) \(11\) \(13\) \(14\) \(16\)
\(\chi_{6003}(34,\cdot)\) \(-1\) \(1\) \(e\left(\frac{125}{462}\right)\) \(e\left(\frac{125}{231}\right)\) \(e\left(\frac{13}{462}\right)\) \(e\left(\frac{401}{462}\right)\) \(e\left(\frac{125}{154}\right)\) \(e\left(\frac{23}{77}\right)\) \(e\left(\frac{229}{231}\right)\) \(e\left(\frac{47}{231}\right)\) \(e\left(\frac{32}{231}\right)\) \(e\left(\frac{19}{231}\right)\)
\(\chi_{6003}(67,\cdot)\) \(-1\) \(1\) \(e\left(\frac{403}{462}\right)\) \(e\left(\frac{172}{231}\right)\) \(e\left(\frac{53}{462}\right)\) \(e\left(\frac{391}{462}\right)\) \(e\left(\frac{95}{154}\right)\) \(e\left(\frac{76}{77}\right)\) \(e\left(\frac{134}{231}\right)\) \(e\left(\frac{85}{231}\right)\) \(e\left(\frac{166}{231}\right)\) \(e\left(\frac{113}{231}\right)\)
\(\chi_{6003}(178,\cdot)\) \(-1\) \(1\) \(e\left(\frac{173}{462}\right)\) \(e\left(\frac{173}{231}\right)\) \(e\left(\frac{103}{462}\right)\) \(e\left(\frac{263}{462}\right)\) \(e\left(\frac{19}{154}\right)\) \(e\left(\frac{46}{77}\right)\) \(e\left(\frac{73}{231}\right)\) \(e\left(\frac{17}{231}\right)\) \(e\left(\frac{218}{231}\right)\) \(e\left(\frac{115}{231}\right)\)
\(\chi_{6003}(241,\cdot)\) \(-1\) \(1\) \(e\left(\frac{389}{462}\right)\) \(e\left(\frac{158}{231}\right)\) \(e\left(\frac{277}{462}\right)\) \(e\left(\frac{335}{462}\right)\) \(e\left(\frac{81}{154}\right)\) \(e\left(\frac{34}{77}\right)\) \(e\left(\frac{64}{231}\right)\) \(e\left(\frac{113}{231}\right)\) \(e\left(\frac{131}{231}\right)\) \(e\left(\frac{85}{231}\right)\)
\(\chi_{6003}(274,\cdot)\) \(-1\) \(1\) \(e\left(\frac{73}{462}\right)\) \(e\left(\frac{73}{231}\right)\) \(e\left(\frac{185}{462}\right)\) \(e\left(\frac{127}{462}\right)\) \(e\left(\frac{73}{154}\right)\) \(e\left(\frac{43}{77}\right)\) \(e\left(\frac{167}{231}\right)\) \(e\left(\frac{118}{231}\right)\) \(e\left(\frac{100}{231}\right)\) \(e\left(\frac{146}{231}\right)\)
\(\chi_{6003}(283,\cdot)\) \(-1\) \(1\) \(e\left(\frac{457}{462}\right)\) \(e\left(\frac{226}{231}\right)\) \(e\left(\frac{443}{462}\right)\) \(e\left(\frac{409}{462}\right)\) \(e\left(\frac{149}{154}\right)\) \(e\left(\frac{73}{77}\right)\) \(e\left(\frac{74}{231}\right)\) \(e\left(\frac{109}{231}\right)\) \(e\left(\frac{202}{231}\right)\) \(e\left(\frac{221}{231}\right)\)
\(\chi_{6003}(295,\cdot)\) \(-1\) \(1\) \(e\left(\frac{377}{462}\right)\) \(e\left(\frac{146}{231}\right)\) \(e\left(\frac{139}{462}\right)\) \(e\left(\frac{23}{462}\right)\) \(e\left(\frac{69}{154}\right)\) \(e\left(\frac{9}{77}\right)\) \(e\left(\frac{103}{231}\right)\) \(e\left(\frac{5}{231}\right)\) \(e\left(\frac{200}{231}\right)\) \(e\left(\frac{61}{231}\right)\)
\(\chi_{6003}(382,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{462}\right)\) \(e\left(\frac{13}{231}\right)\) \(e\left(\frac{419}{462}\right)\) \(e\left(\frac{415}{462}\right)\) \(e\left(\frac{13}{154}\right)\) \(e\left(\frac{72}{77}\right)\) \(e\left(\frac{131}{231}\right)\) \(e\left(\frac{40}{231}\right)\) \(e\left(\frac{214}{231}\right)\) \(e\left(\frac{26}{231}\right)\)
\(\chi_{6003}(412,\cdot)\) \(-1\) \(1\) \(e\left(\frac{29}{462}\right)\) \(e\left(\frac{29}{231}\right)\) \(e\left(\frac{295}{462}\right)\) \(e\left(\frac{215}{462}\right)\) \(e\left(\frac{29}{154}\right)\) \(e\left(\frac{54}{77}\right)\) \(e\left(\frac{79}{231}\right)\) \(e\left(\frac{107}{231}\right)\) \(e\left(\frac{122}{231}\right)\) \(e\left(\frac{58}{231}\right)\)
\(\chi_{6003}(448,\cdot)\) \(-1\) \(1\) \(e\left(\frac{59}{462}\right)\) \(e\left(\frac{59}{231}\right)\) \(e\left(\frac{409}{462}\right)\) \(e\left(\frac{71}{462}\right)\) \(e\left(\frac{59}{154}\right)\) \(e\left(\frac{1}{77}\right)\) \(e\left(\frac{97}{231}\right)\) \(e\left(\frac{146}{231}\right)\) \(e\left(\frac{65}{231}\right)\) \(e\left(\frac{118}{231}\right)\)
\(\chi_{6003}(457,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{462}\right)\) \(e\left(\frac{23}{231}\right)\) \(e\left(\frac{457}{462}\right)\) \(e\left(\frac{59}{462}\right)\) \(e\left(\frac{23}{154}\right)\) \(e\left(\frac{3}{77}\right)\) \(e\left(\frac{214}{231}\right)\) \(e\left(\frac{53}{231}\right)\) \(e\left(\frac{41}{231}\right)\) \(e\left(\frac{46}{231}\right)\)
\(\chi_{6003}(502,\cdot)\) \(-1\) \(1\) \(e\left(\frac{179}{462}\right)\) \(e\left(\frac{179}{231}\right)\) \(e\left(\frac{403}{462}\right)\) \(e\left(\frac{419}{462}\right)\) \(e\left(\frac{25}{154}\right)\) \(e\left(\frac{20}{77}\right)\) \(e\left(\frac{169}{231}\right)\) \(e\left(\frac{71}{231}\right)\) \(e\left(\frac{68}{231}\right)\) \(e\left(\frac{127}{231}\right)\)
\(\chi_{6003}(526,\cdot)\) \(-1\) \(1\) \(e\left(\frac{397}{462}\right)\) \(e\left(\frac{166}{231}\right)\) \(e\left(\frac{215}{462}\right)\) \(e\left(\frac{235}{462}\right)\) \(e\left(\frac{89}{154}\right)\) \(e\left(\frac{25}{77}\right)\) \(e\left(\frac{38}{231}\right)\) \(e\left(\frac{31}{231}\right)\) \(e\left(\frac{85}{231}\right)\) \(e\left(\frac{101}{231}\right)\)
\(\chi_{6003}(544,\cdot)\) \(-1\) \(1\) \(e\left(\frac{373}{462}\right)\) \(e\left(\frac{142}{231}\right)\) \(e\left(\frac{401}{462}\right)\) \(e\left(\frac{73}{462}\right)\) \(e\left(\frac{65}{154}\right)\) \(e\left(\frac{52}{77}\right)\) \(e\left(\frac{116}{231}\right)\) \(e\left(\frac{46}{231}\right)\) \(e\left(\frac{223}{231}\right)\) \(e\left(\frac{53}{231}\right)\)
\(\chi_{6003}(589,\cdot)\) \(-1\) \(1\) \(e\left(\frac{277}{462}\right)\) \(e\left(\frac{46}{231}\right)\) \(e\left(\frac{221}{462}\right)\) \(e\left(\frac{349}{462}\right)\) \(e\left(\frac{123}{154}\right)\) \(e\left(\frac{6}{77}\right)\) \(e\left(\frac{197}{231}\right)\) \(e\left(\frac{106}{231}\right)\) \(e\left(\frac{82}{231}\right)\) \(e\left(\frac{92}{231}\right)\)
\(\chi_{6003}(700,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{462}\right)\) \(e\left(\frac{5}{231}\right)\) \(e\left(\frac{19}{462}\right)\) \(e\left(\frac{53}{462}\right)\) \(e\left(\frac{5}{154}\right)\) \(e\left(\frac{4}{77}\right)\) \(e\left(\frac{157}{231}\right)\) \(e\left(\frac{122}{231}\right)\) \(e\left(\frac{29}{231}\right)\) \(e\left(\frac{10}{231}\right)\)
\(\chi_{6003}(709,\cdot)\) \(-1\) \(1\) \(e\left(\frac{311}{462}\right)\) \(e\left(\frac{80}{231}\right)\) \(e\left(\frac{73}{462}\right)\) \(e\left(\frac{155}{462}\right)\) \(e\left(\frac{3}{154}\right)\) \(e\left(\frac{64}{77}\right)\) \(e\left(\frac{202}{231}\right)\) \(e\left(\frac{104}{231}\right)\) \(e\left(\frac{2}{231}\right)\) \(e\left(\frac{160}{231}\right)\)
\(\chi_{6003}(718,\cdot)\) \(-1\) \(1\) \(e\left(\frac{317}{462}\right)\) \(e\left(\frac{86}{231}\right)\) \(e\left(\frac{373}{462}\right)\) \(e\left(\frac{311}{462}\right)\) \(e\left(\frac{9}{154}\right)\) \(e\left(\frac{38}{77}\right)\) \(e\left(\frac{67}{231}\right)\) \(e\left(\frac{158}{231}\right)\) \(e\left(\frac{83}{231}\right)\) \(e\left(\frac{172}{231}\right)\)
\(\chi_{6003}(787,\cdot)\) \(-1\) \(1\) \(e\left(\frac{229}{462}\right)\) \(e\left(\frac{229}{231}\right)\) \(e\left(\frac{131}{462}\right)\) \(e\left(\frac{25}{462}\right)\) \(e\left(\frac{75}{154}\right)\) \(e\left(\frac{60}{77}\right)\) \(e\left(\frac{122}{231}\right)\) \(e\left(\frac{136}{231}\right)\) \(e\left(\frac{127}{231}\right)\) \(e\left(\frac{227}{231}\right)\)
\(\chi_{6003}(796,\cdot)\) \(-1\) \(1\) \(e\left(\frac{409}{462}\right)\) \(e\left(\frac{178}{231}\right)\) \(e\left(\frac{353}{462}\right)\) \(e\left(\frac{85}{462}\right)\) \(e\left(\frac{101}{154}\right)\) \(e\left(\frac{50}{77}\right)\) \(e\left(\frac{230}{231}\right)\) \(e\left(\frac{139}{231}\right)\) \(e\left(\frac{16}{231}\right)\) \(e\left(\frac{125}{231}\right)\)
\(\chi_{6003}(904,\cdot)\) \(-1\) \(1\) \(e\left(\frac{391}{462}\right)\) \(e\left(\frac{160}{231}\right)\) \(e\left(\frac{377}{462}\right)\) \(e\left(\frac{79}{462}\right)\) \(e\left(\frac{83}{154}\right)\) \(e\left(\frac{51}{77}\right)\) \(e\left(\frac{173}{231}\right)\) \(e\left(\frac{208}{231}\right)\) \(e\left(\frac{4}{231}\right)\) \(e\left(\frac{89}{231}\right)\)
\(\chi_{6003}(934,\cdot)\) \(-1\) \(1\) \(e\left(\frac{365}{462}\right)\) \(e\left(\frac{134}{231}\right)\) \(e\left(\frac{1}{462}\right)\) \(e\left(\frac{173}{462}\right)\) \(e\left(\frac{57}{154}\right)\) \(e\left(\frac{61}{77}\right)\) \(e\left(\frac{142}{231}\right)\) \(e\left(\frac{128}{231}\right)\) \(e\left(\frac{38}{231}\right)\) \(e\left(\frac{37}{231}\right)\)
\(\chi_{6003}(1078,\cdot)\) \(-1\) \(1\) \(e\left(\frac{419}{462}\right)\) \(e\left(\frac{188}{231}\right)\) \(e\left(\frac{391}{462}\right)\) \(e\left(\frac{191}{462}\right)\) \(e\left(\frac{111}{154}\right)\) \(e\left(\frac{58}{77}\right)\) \(e\left(\frac{82}{231}\right)\) \(e\left(\frac{152}{231}\right)\) \(e\left(\frac{74}{231}\right)\) \(e\left(\frac{145}{231}\right)\)
\(\chi_{6003}(1111,\cdot)\) \(-1\) \(1\) \(e\left(\frac{193}{462}\right)\) \(e\left(\frac{193}{231}\right)\) \(e\left(\frac{179}{462}\right)\) \(e\left(\frac{13}{462}\right)\) \(e\left(\frac{39}{154}\right)\) \(e\left(\frac{62}{77}\right)\) \(e\left(\frac{8}{231}\right)\) \(e\left(\frac{43}{231}\right)\) \(e\left(\frac{103}{231}\right)\) \(e\left(\frac{155}{231}\right)\)
\(\chi_{6003}(1165,\cdot)\) \(-1\) \(1\) \(e\left(\frac{307}{462}\right)\) \(e\left(\frac{76}{231}\right)\) \(e\left(\frac{335}{462}\right)\) \(e\left(\frac{205}{462}\right)\) \(e\left(\frac{153}{154}\right)\) \(e\left(\frac{30}{77}\right)\) \(e\left(\frac{215}{231}\right)\) \(e\left(\frac{145}{231}\right)\) \(e\left(\frac{25}{231}\right)\) \(e\left(\frac{152}{231}\right)\)
\(\chi_{6003}(1240,\cdot)\) \(-1\) \(1\) \(e\left(\frac{359}{462}\right)\) \(e\left(\frac{128}{231}\right)\) \(e\left(\frac{163}{462}\right)\) \(e\left(\frac{17}{462}\right)\) \(e\left(\frac{51}{154}\right)\) \(e\left(\frac{10}{77}\right)\) \(e\left(\frac{46}{231}\right)\) \(e\left(\frac{74}{231}\right)\) \(e\left(\frac{188}{231}\right)\) \(e\left(\frac{25}{231}\right)\)
\(\chi_{6003}(1282,\cdot)\) \(-1\) \(1\) \(e\left(\frac{85}{462}\right)\) \(e\left(\frac{85}{231}\right)\) \(e\left(\frac{323}{462}\right)\) \(e\left(\frac{439}{462}\right)\) \(e\left(\frac{85}{154}\right)\) \(e\left(\frac{68}{77}\right)\) \(e\left(\frac{128}{231}\right)\) \(e\left(\frac{226}{231}\right)\) \(e\left(\frac{31}{231}\right)\) \(e\left(\frac{170}{231}\right)\)
\(\chi_{6003}(1285,\cdot)\) \(-1\) \(1\) \(e\left(\frac{221}{462}\right)\) \(e\left(\frac{221}{231}\right)\) \(e\left(\frac{193}{462}\right)\) \(e\left(\frac{125}{462}\right)\) \(e\left(\frac{67}{154}\right)\) \(e\left(\frac{69}{77}\right)\) \(e\left(\frac{148}{231}\right)\) \(e\left(\frac{218}{231}\right)\) \(e\left(\frac{173}{231}\right)\) \(e\left(\frac{211}{231}\right)\)
\(\chi_{6003}(1309,\cdot)\) \(-1\) \(1\) \(e\left(\frac{271}{462}\right)\) \(e\left(\frac{40}{231}\right)\) \(e\left(\frac{383}{462}\right)\) \(e\left(\frac{193}{462}\right)\) \(e\left(\frac{117}{154}\right)\) \(e\left(\frac{32}{77}\right)\) \(e\left(\frac{101}{231}\right)\) \(e\left(\frac{52}{231}\right)\) \(e\left(\frac{1}{231}\right)\) \(e\left(\frac{80}{231}\right)\)
\(\chi_{6003}(1318,\cdot)\) \(-1\) \(1\) \(e\left(\frac{325}{462}\right)\) \(e\left(\frac{94}{231}\right)\) \(e\left(\frac{311}{462}\right)\) \(e\left(\frac{211}{462}\right)\) \(e\left(\frac{17}{154}\right)\) \(e\left(\frac{29}{77}\right)\) \(e\left(\frac{41}{231}\right)\) \(e\left(\frac{76}{231}\right)\) \(e\left(\frac{37}{231}\right)\) \(e\left(\frac{188}{231}\right)\)
\(\chi_{6003}(1339,\cdot)\) \(-1\) \(1\) \(e\left(\frac{251}{462}\right)\) \(e\left(\frac{20}{231}\right)\) \(e\left(\frac{307}{462}\right)\) \(e\left(\frac{443}{462}\right)\) \(e\left(\frac{97}{154}\right)\) \(e\left(\frac{16}{77}\right)\) \(e\left(\frac{166}{231}\right)\) \(e\left(\frac{26}{231}\right)\) \(e\left(\frac{116}{231}\right)\) \(e\left(\frac{40}{231}\right)\)