from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6003, base_ring=CyclotomicField(462))
M = H._module
chi = DirichletCharacter(H, M([154,84,33]))
chi.galois_orbit()
[g,chi] = znchar(Mod(4,6003))
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: | even | 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 28 of 120 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{6003}(4,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{355}{462}\right)\) | \(e\left(\frac{124}{231}\right)\) | \(e\left(\frac{97}{231}\right)\) | \(e\left(\frac{149}{231}\right)\) | \(e\left(\frac{47}{154}\right)\) | \(e\left(\frac{29}{154}\right)\) | \(e\left(\frac{349}{462}\right)\) | \(e\left(\frac{115}{231}\right)\) | \(e\left(\frac{191}{462}\right)\) | \(e\left(\frac{17}{231}\right)\) |
| \(\chi_{6003}(13,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{115}{462}\right)\) | \(e\left(\frac{115}{231}\right)\) | \(e\left(\frac{103}{231}\right)\) | \(e\left(\frac{32}{231}\right)\) | \(e\left(\frac{115}{154}\right)\) | \(e\left(\frac{107}{154}\right)\) | \(e\left(\frac{61}{462}\right)\) | \(e\left(\frac{34}{231}\right)\) | \(e\left(\frac{179}{462}\right)\) | \(e\left(\frac{230}{231}\right)\) |
| \(\chi_{6003}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{349}{462}\right)\) | \(e\left(\frac{118}{231}\right)\) | \(e\left(\frac{178}{231}\right)\) | \(e\left(\frac{71}{231}\right)\) | \(e\left(\frac{41}{154}\right)\) | \(e\left(\frac{81}{154}\right)\) | \(e\left(\frac{157}{462}\right)\) | \(e\left(\frac{61}{231}\right)\) | \(e\left(\frac{29}{462}\right)\) | \(e\left(\frac{5}{231}\right)\) |
| \(\chi_{6003}(151,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{462}\right)\) | \(e\left(\frac{71}{231}\right)\) | \(e\left(\frac{158}{231}\right)\) | \(e\left(\frac{76}{231}\right)\) | \(e\left(\frac{71}{154}\right)\) | \(e\left(\frac{129}{154}\right)\) | \(e\left(\frac{347}{462}\right)\) | \(e\left(\frac{23}{231}\right)\) | \(e\left(\frac{223}{462}\right)\) | \(e\left(\frac{142}{231}\right)\) |
| \(\chi_{6003}(187,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{353}{462}\right)\) | \(e\left(\frac{122}{231}\right)\) | \(e\left(\frac{47}{231}\right)\) | \(e\left(\frac{46}{231}\right)\) | \(e\left(\frac{45}{154}\right)\) | \(e\left(\frac{149}{154}\right)\) | \(e\left(\frac{131}{462}\right)\) | \(e\left(\frac{20}{231}\right)\) | \(e\left(\frac{445}{462}\right)\) | \(e\left(\frac{13}{231}\right)\) |
| \(\chi_{6003}(196,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{191}{462}\right)\) | \(e\left(\frac{191}{231}\right)\) | \(e\left(\frac{155}{231}\right)\) | \(e\left(\frac{19}{231}\right)\) | \(e\left(\frac{37}{154}\right)\) | \(e\left(\frac{13}{154}\right)\) | \(e\left(\frac{29}{462}\right)\) | \(e\left(\frac{179}{231}\right)\) | \(e\left(\frac{229}{462}\right)\) | \(e\left(\frac{151}{231}\right)\) |
| \(\chi_{6003}(238,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{462}\right)\) | \(e\left(\frac{43}{231}\right)\) | \(e\left(\frac{151}{231}\right)\) | \(e\left(\frac{20}{231}\right)\) | \(e\left(\frac{43}{154}\right)\) | \(e\left(\frac{115}{154}\right)\) | \(e\left(\frac{67}{462}\right)\) | \(e\left(\frac{79}{231}\right)\) | \(e\left(\frac{83}{462}\right)\) | \(e\left(\frac{86}{231}\right)\) |
| \(\chi_{6003}(265,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{462}\right)\) | \(e\left(\frac{103}{231}\right)\) | \(e\left(\frac{34}{231}\right)\) | \(e\left(\frac{107}{231}\right)\) | \(e\left(\frac{103}{154}\right)\) | \(e\left(\frac{57}{154}\right)\) | \(e\left(\frac{139}{462}\right)\) | \(e\left(\frac{157}{231}\right)\) | \(e\left(\frac{317}{462}\right)\) | \(e\left(\frac{206}{231}\right)\) |
| \(\chi_{6003}(328,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{151}{462}\right)\) | \(e\left(\frac{151}{231}\right)\) | \(e\left(\frac{79}{231}\right)\) | \(e\left(\frac{38}{231}\right)\) | \(e\left(\frac{151}{154}\right)\) | \(e\left(\frac{103}{154}\right)\) | \(e\left(\frac{289}{462}\right)\) | \(e\left(\frac{127}{231}\right)\) | \(e\left(\frac{227}{462}\right)\) | \(e\left(\frac{71}{231}\right)\) |
| \(\chi_{6003}(439,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{425}{462}\right)\) | \(e\left(\frac{194}{231}\right)\) | \(e\left(\frac{230}{231}\right)\) | \(e\left(\frac{58}{231}\right)\) | \(e\left(\frac{117}{154}\right)\) | \(e\left(\frac{141}{154}\right)\) | \(e\left(\frac{125}{462}\right)\) | \(e\left(\frac{206}{231}\right)\) | \(e\left(\frac{79}{462}\right)\) | \(e\left(\frac{157}{231}\right)\) |
| \(\chi_{6003}(499,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{127}{462}\right)\) | \(e\left(\frac{127}{231}\right)\) | \(e\left(\frac{172}{231}\right)\) | \(e\left(\frac{188}{231}\right)\) | \(e\left(\frac{127}{154}\right)\) | \(e\left(\frac{3}{154}\right)\) | \(e\left(\frac{445}{462}\right)\) | \(e\left(\frac{142}{231}\right)\) | \(e\left(\frac{41}{462}\right)\) | \(e\left(\frac{23}{231}\right)\) |
| \(\chi_{6003}(535,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{283}{462}\right)\) | \(e\left(\frac{52}{231}\right)\) | \(e\left(\frac{145}{231}\right)\) | \(e\left(\frac{137}{231}\right)\) | \(e\left(\frac{129}{154}\right)\) | \(e\left(\frac{37}{154}\right)\) | \(e\left(\frac{355}{462}\right)\) | \(e\left(\frac{160}{231}\right)\) | \(e\left(\frac{95}{462}\right)\) | \(e\left(\frac{104}{231}\right)\) |
| \(\chi_{6003}(556,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{377}{462}\right)\) | \(e\left(\frac{146}{231}\right)\) | \(e\left(\frac{185}{231}\right)\) | \(e\left(\frac{127}{231}\right)\) | \(e\left(\frac{69}{154}\right)\) | \(e\left(\frac{95}{154}\right)\) | \(e\left(\frac{437}{462}\right)\) | \(e\left(\frac{5}{231}\right)\) | \(e\left(\frac{169}{462}\right)\) | \(e\left(\frac{61}{231}\right)\) |
| \(\chi_{6003}(673,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{239}{462}\right)\) | \(e\left(\frac{8}{231}\right)\) | \(e\left(\frac{200}{231}\right)\) | \(e\left(\frac{181}{231}\right)\) | \(e\left(\frac{85}{154}\right)\) | \(e\left(\frac{59}{154}\right)\) | \(e\left(\frac{179}{462}\right)\) | \(e\left(\frac{149}{231}\right)\) | \(e\left(\frac{139}{462}\right)\) | \(e\left(\frac{16}{231}\right)\) |
| \(\chi_{6003}(763,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{179}{462}\right)\) | \(e\left(\frac{179}{231}\right)\) | \(e\left(\frac{86}{231}\right)\) | \(e\left(\frac{94}{231}\right)\) | \(e\left(\frac{25}{154}\right)\) | \(e\left(\frac{117}{154}\right)\) | \(e\left(\frac{107}{462}\right)\) | \(e\left(\frac{71}{231}\right)\) | \(e\left(\frac{367}{462}\right)\) | \(e\left(\frac{127}{231}\right)\) |
| \(\chi_{6003}(817,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{125}{462}\right)\) | \(e\left(\frac{125}{231}\right)\) | \(e\left(\frac{122}{231}\right)\) | \(e\left(\frac{85}{231}\right)\) | \(e\left(\frac{125}{154}\right)\) | \(e\left(\frac{123}{154}\right)\) | \(e\left(\frac{227}{462}\right)\) | \(e\left(\frac{47}{231}\right)\) | \(e\left(\frac{295}{462}\right)\) | \(e\left(\frac{19}{231}\right)\) |
| \(\chi_{6003}(961,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{383}{462}\right)\) | \(e\left(\frac{152}{231}\right)\) | \(e\left(\frac{104}{231}\right)\) | \(e\left(\frac{205}{231}\right)\) | \(e\left(\frac{75}{154}\right)\) | \(e\left(\frac{43}{154}\right)\) | \(e\left(\frac{167}{462}\right)\) | \(e\left(\frac{59}{231}\right)\) | \(e\left(\frac{331}{462}\right)\) | \(e\left(\frac{73}{231}\right)\) |
| \(\chi_{6003}(970,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{311}{462}\right)\) | \(e\left(\frac{80}{231}\right)\) | \(e\left(\frac{152}{231}\right)\) | \(e\left(\frac{193}{231}\right)\) | \(e\left(\frac{3}{154}\right)\) | \(e\left(\frac{51}{154}\right)\) | \(e\left(\frac{173}{462}\right)\) | \(e\left(\frac{104}{231}\right)\) | \(e\left(\frac{235}{462}\right)\) | \(e\left(\frac{160}{231}\right)\) |
| \(\chi_{6003}(979,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{401}{462}\right)\) | \(e\left(\frac{170}{231}\right)\) | \(e\left(\frac{92}{231}\right)\) | \(e\left(\frac{208}{231}\right)\) | \(e\left(\frac{93}{154}\right)\) | \(e\left(\frac{41}{154}\right)\) | \(e\left(\frac{281}{462}\right)\) | \(e\left(\frac{221}{231}\right)\) | \(e\left(\frac{355}{462}\right)\) | \(e\left(\frac{109}{231}\right)\) |
| \(\chi_{6003}(1021,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{211}{462}\right)\) | \(e\left(\frac{211}{231}\right)\) | \(e\left(\frac{193}{231}\right)\) | \(e\left(\frac{125}{231}\right)\) | \(e\left(\frac{57}{154}\right)\) | \(e\left(\frac{45}{154}\right)\) | \(e\left(\frac{361}{462}\right)\) | \(e\left(\frac{205}{231}\right)\) | \(e\left(\frac{461}{462}\right)\) | \(e\left(\frac{191}{231}\right)\) |
| \(\chi_{6003}(1024,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{389}{462}\right)\) | \(e\left(\frac{158}{231}\right)\) | \(e\left(\frac{23}{231}\right)\) | \(e\left(\frac{52}{231}\right)\) | \(e\left(\frac{81}{154}\right)\) | \(e\left(\frac{145}{154}\right)\) | \(e\left(\frac{359}{462}\right)\) | \(e\left(\frac{113}{231}\right)\) | \(e\left(\frac{31}{462}\right)\) | \(e\left(\frac{85}{231}\right)\) |
| \(\chi_{6003}(1048,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{313}{462}\right)\) | \(e\left(\frac{82}{231}\right)\) | \(e\left(\frac{202}{231}\right)\) | \(e\left(\frac{65}{231}\right)\) | \(e\left(\frac{5}{154}\right)\) | \(e\left(\frac{85}{154}\right)\) | \(e\left(\frac{391}{462}\right)\) | \(e\left(\frac{199}{231}\right)\) | \(e\left(\frac{443}{462}\right)\) | \(e\left(\frac{164}{231}\right)\) |
| \(\chi_{6003}(1066,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{373}{462}\right)\) | \(e\left(\frac{142}{231}\right)\) | \(e\left(\frac{85}{231}\right)\) | \(e\left(\frac{152}{231}\right)\) | \(e\left(\frac{65}{154}\right)\) | \(e\left(\frac{27}{154}\right)\) | \(e\left(\frac{1}{462}\right)\) | \(e\left(\frac{46}{231}\right)\) | \(e\left(\frac{215}{462}\right)\) | \(e\left(\frac{53}{231}\right)\) |
| \(\chi_{6003}(1222,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{462}\right)\) | \(e\left(\frac{89}{231}\right)\) | \(e\left(\frac{146}{231}\right)\) | \(e\left(\frac{79}{231}\right)\) | \(e\left(\frac{89}{154}\right)\) | \(e\left(\frac{127}{154}\right)\) | \(e\left(\frac{461}{462}\right)\) | \(e\left(\frac{185}{231}\right)\) | \(e\left(\frac{247}{462}\right)\) | \(e\left(\frac{178}{231}\right)\) |
| \(\chi_{6003}(1231,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{462}\right)\) | \(e\left(\frac{59}{231}\right)\) | \(e\left(\frac{89}{231}\right)\) | \(e\left(\frac{151}{231}\right)\) | \(e\left(\frac{59}{154}\right)\) | \(e\left(\frac{79}{154}\right)\) | \(e\left(\frac{425}{462}\right)\) | \(e\left(\frac{146}{231}\right)\) | \(e\left(\frac{361}{462}\right)\) | \(e\left(\frac{118}{231}\right)\) |
| \(\chi_{6003}(1327,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{457}{462}\right)\) | \(e\left(\frac{226}{231}\right)\) | \(e\left(\frac{106}{231}\right)\) | \(e\left(\frac{89}{231}\right)\) | \(e\left(\frac{149}{154}\right)\) | \(e\left(\frac{69}{154}\right)\) | \(e\left(\frac{379}{462}\right)\) | \(e\left(\frac{109}{231}\right)\) | \(e\left(\frac{173}{462}\right)\) | \(e\left(\frac{221}{231}\right)\) |
| \(\chi_{6003}(1501,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{462}\right)\) | \(e\left(\frac{107}{231}\right)\) | \(e\left(\frac{134}{231}\right)\) | \(e\left(\frac{82}{231}\right)\) | \(e\left(\frac{107}{154}\right)\) | \(e\left(\frac{125}{154}\right)\) | \(e\left(\frac{113}{462}\right)\) | \(e\left(\frac{116}{231}\right)\) | \(e\left(\frac{271}{462}\right)\) | \(e\left(\frac{214}{231}\right)\) |
| \(\chi_{6003}(1543,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{337}{462}\right)\) | \(e\left(\frac{106}{231}\right)\) | \(e\left(\frac{109}{231}\right)\) | \(e\left(\frac{146}{231}\right)\) | \(e\left(\frac{29}{154}\right)\) | \(e\left(\frac{31}{154}\right)\) | \(e\left(\frac{235}{462}\right)\) | \(e\left(\frac{184}{231}\right)\) | \(e\left(\frac{167}{462}\right)\) | \(e\left(\frac{212}{231}\right)\) |