from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6003, base_ring=CyclotomicField(462))
M = H._module
chi = DirichletCharacter(H, M([385,21,363]))
chi.galois_orbit()
[g,chi] = znchar(Mod(5,6003))
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 30 of 120 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{6003}(5,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{164}{231}\right)\) | \(e\left(\frac{97}{231}\right)\) | \(e\left(\frac{115}{231}\right)\) | \(e\left(\frac{289}{462}\right)\) | \(e\left(\frac{10}{77}\right)\) | \(e\left(\frac{16}{77}\right)\) | \(e\left(\frac{409}{462}\right)\) | \(e\left(\frac{103}{231}\right)\) | \(e\left(\frac{155}{462}\right)\) | \(e\left(\frac{194}{231}\right)\) |
| \(\chi_{6003}(38,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{231}\right)\) | \(e\left(\frac{32}{231}\right)\) | \(e\left(\frac{107}{231}\right)\) | \(e\left(\frac{293}{462}\right)\) | \(e\left(\frac{16}{77}\right)\) | \(e\left(\frac{41}{77}\right)\) | \(e\left(\frac{23}{462}\right)\) | \(e\left(\frac{134}{231}\right)\) | \(e\left(\frac{325}{462}\right)\) | \(e\left(\frac{64}{231}\right)\) |
| \(\chi_{6003}(122,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{179}{231}\right)\) | \(e\left(\frac{127}{231}\right)\) | \(e\left(\frac{172}{231}\right)\) | \(e\left(\frac{145}{462}\right)\) | \(e\left(\frac{25}{77}\right)\) | \(e\left(\frac{40}{77}\right)\) | \(e\left(\frac{445}{462}\right)\) | \(e\left(\frac{142}{231}\right)\) | \(e\left(\frac{41}{462}\right)\) | \(e\left(\frac{23}{231}\right)\) |
| \(\chi_{6003}(149,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{167}{231}\right)\) | \(e\left(\frac{103}{231}\right)\) | \(e\left(\frac{34}{231}\right)\) | \(e\left(\frac{445}{462}\right)\) | \(e\left(\frac{13}{77}\right)\) | \(e\left(\frac{67}{77}\right)\) | \(e\left(\frac{139}{462}\right)\) | \(e\left(\frac{157}{231}\right)\) | \(e\left(\frac{317}{462}\right)\) | \(e\left(\frac{206}{231}\right)\) |
| \(\chi_{6003}(158,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{215}{231}\right)\) | \(e\left(\frac{199}{231}\right)\) | \(e\left(\frac{124}{231}\right)\) | \(e\left(\frac{169}{462}\right)\) | \(e\left(\frac{61}{77}\right)\) | \(e\left(\frac{36}{77}\right)\) | \(e\left(\frac{439}{462}\right)\) | \(e\left(\frac{97}{231}\right)\) | \(e\left(\frac{137}{462}\right)\) | \(e\left(\frac{167}{231}\right)\) |
| \(\chi_{6003}(212,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{65}{231}\right)\) | \(e\left(\frac{130}{231}\right)\) | \(e\left(\frac{16}{231}\right)\) | \(e\left(\frac{223}{462}\right)\) | \(e\left(\frac{65}{77}\right)\) | \(e\left(\frac{27}{77}\right)\) | \(e\left(\frac{79}{462}\right)\) | \(e\left(\frac{169}{231}\right)\) | \(e\left(\frac{353}{462}\right)\) | \(e\left(\frac{29}{231}\right)\) |
| \(\chi_{6003}(245,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{82}{231}\right)\) | \(e\left(\frac{164}{231}\right)\) | \(e\left(\frac{173}{231}\right)\) | \(e\left(\frac{29}{462}\right)\) | \(e\left(\frac{5}{77}\right)\) | \(e\left(\frac{8}{77}\right)\) | \(e\left(\frac{89}{462}\right)\) | \(e\left(\frac{167}{231}\right)\) | \(e\left(\frac{193}{462}\right)\) | \(e\left(\frac{97}{231}\right)\) |
| \(\chi_{6003}(383,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{137}{231}\right)\) | \(e\left(\frac{43}{231}\right)\) | \(e\left(\frac{151}{231}\right)\) | \(e\left(\frac{271}{462}\right)\) | \(e\left(\frac{60}{77}\right)\) | \(e\left(\frac{19}{77}\right)\) | \(e\left(\frac{67}{462}\right)\) | \(e\left(\frac{79}{231}\right)\) | \(e\left(\frac{83}{462}\right)\) | \(e\left(\frac{86}{231}\right)\) |
| \(\chi_{6003}(410,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{62}{231}\right)\) | \(e\left(\frac{124}{231}\right)\) | \(e\left(\frac{97}{231}\right)\) | \(e\left(\frac{67}{462}\right)\) | \(e\left(\frac{62}{77}\right)\) | \(e\left(\frac{53}{77}\right)\) | \(e\left(\frac{349}{462}\right)\) | \(e\left(\frac{115}{231}\right)\) | \(e\left(\frac{191}{462}\right)\) | \(e\left(\frac{17}{231}\right)\) |
| \(\chi_{6003}(419,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{131}{231}\right)\) | \(e\left(\frac{31}{231}\right)\) | \(e\left(\frac{82}{231}\right)\) | \(e\left(\frac{421}{462}\right)\) | \(e\left(\frac{54}{77}\right)\) | \(e\left(\frac{71}{77}\right)\) | \(e\left(\frac{145}{462}\right)\) | \(e\left(\frac{202}{231}\right)\) | \(e\left(\frac{221}{462}\right)\) | \(e\left(\frac{62}{231}\right)\) |
| \(\chi_{6003}(428,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{155}{231}\right)\) | \(e\left(\frac{79}{231}\right)\) | \(e\left(\frac{127}{231}\right)\) | \(e\left(\frac{283}{462}\right)\) | \(e\left(\frac{1}{77}\right)\) | \(e\left(\frac{17}{77}\right)\) | \(e\left(\frac{295}{462}\right)\) | \(e\left(\frac{172}{231}\right)\) | \(e\left(\frac{131}{462}\right)\) | \(e\left(\frac{158}{231}\right)\) |
| \(\chi_{6003}(470,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{151}{231}\right)\) | \(e\left(\frac{71}{231}\right)\) | \(e\left(\frac{158}{231}\right)\) | \(e\left(\frac{383}{462}\right)\) | \(e\left(\frac{74}{77}\right)\) | \(e\left(\frac{26}{77}\right)\) | \(e\left(\frac{347}{462}\right)\) | \(e\left(\frac{23}{231}\right)\) | \(e\left(\frac{223}{462}\right)\) | \(e\left(\frac{142}{231}\right)\) |
| \(\chi_{6003}(497,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{231}\right)\) | \(e\left(\frac{68}{231}\right)\) | \(e\left(\frac{83}{231}\right)\) | \(e\left(\frac{305}{462}\right)\) | \(e\left(\frac{34}{77}\right)\) | \(e\left(\frac{39}{77}\right)\) | \(e\left(\frac{251}{462}\right)\) | \(e\left(\frac{227}{231}\right)\) | \(e\left(\frac{373}{462}\right)\) | \(e\left(\frac{136}{231}\right)\) |
| \(\chi_{6003}(527,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{185}{231}\right)\) | \(e\left(\frac{139}{231}\right)\) | \(e\left(\frac{10}{231}\right)\) | \(e\left(\frac{457}{462}\right)\) | \(e\left(\frac{31}{77}\right)\) | \(e\left(\frac{65}{77}\right)\) | \(e\left(\frac{367}{462}\right)\) | \(e\left(\frac{19}{231}\right)\) | \(e\left(\frac{365}{462}\right)\) | \(e\left(\frac{47}{231}\right)\) |
| \(\chi_{6003}(734,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{86}{231}\right)\) | \(e\left(\frac{172}{231}\right)\) | \(e\left(\frac{142}{231}\right)\) | \(e\left(\frac{391}{462}\right)\) | \(e\left(\frac{9}{77}\right)\) | \(e\left(\frac{76}{77}\right)\) | \(e\left(\frac{37}{462}\right)\) | \(e\left(\frac{85}{231}\right)\) | \(e\left(\frac{101}{462}\right)\) | \(e\left(\frac{113}{231}\right)\) |
| \(\chi_{6003}(776,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{169}{231}\right)\) | \(e\left(\frac{107}{231}\right)\) | \(e\left(\frac{134}{231}\right)\) | \(e\left(\frac{395}{462}\right)\) | \(e\left(\frac{15}{77}\right)\) | \(e\left(\frac{24}{77}\right)\) | \(e\left(\frac{113}{462}\right)\) | \(e\left(\frac{116}{231}\right)\) | \(e\left(\frac{271}{462}\right)\) | \(e\left(\frac{214}{231}\right)\) |
| \(\chi_{6003}(941,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{152}{231}\right)\) | \(e\left(\frac{73}{231}\right)\) | \(e\left(\frac{208}{231}\right)\) | \(e\left(\frac{127}{462}\right)\) | \(e\left(\frac{75}{77}\right)\) | \(e\left(\frac{43}{77}\right)\) | \(e\left(\frac{103}{462}\right)\) | \(e\left(\frac{118}{231}\right)\) | \(e\left(\frac{431}{462}\right)\) | \(e\left(\frac{146}{231}\right)\) |
| \(\chi_{6003}(950,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{231}\right)\) | \(e\left(\frac{226}{231}\right)\) | \(e\left(\frac{106}{231}\right)\) | \(e\left(\frac{409}{462}\right)\) | \(e\left(\frac{36}{77}\right)\) | \(e\left(\frac{73}{77}\right)\) | \(e\left(\frac{379}{462}\right)\) | \(e\left(\frac{109}{231}\right)\) | \(e\left(\frac{173}{462}\right)\) | \(e\left(\frac{221}{231}\right)\) |
| \(\chi_{6003}(1019,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{223}{231}\right)\) | \(e\left(\frac{215}{231}\right)\) | \(e\left(\frac{62}{231}\right)\) | \(e\left(\frac{431}{462}\right)\) | \(e\left(\frac{69}{77}\right)\) | \(e\left(\frac{18}{77}\right)\) | \(e\left(\frac{335}{462}\right)\) | \(e\left(\frac{164}{231}\right)\) | \(e\left(\frac{415}{462}\right)\) | \(e\left(\frac{199}{231}\right)\) |
| \(\chi_{6003}(1049,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{122}{231}\right)\) | \(e\left(\frac{13}{231}\right)\) | \(e\left(\frac{94}{231}\right)\) | \(e\left(\frac{415}{462}\right)\) | \(e\left(\frac{45}{77}\right)\) | \(e\left(\frac{72}{77}\right)\) | \(e\left(\frac{31}{462}\right)\) | \(e\left(\frac{40}{231}\right)\) | \(e\left(\frac{197}{462}\right)\) | \(e\left(\frac{26}{231}\right)\) |
| \(\chi_{6003}(1193,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{231}\right)\) | \(e\left(\frac{166}{231}\right)\) | \(e\left(\frac{223}{231}\right)\) | \(e\left(\frac{235}{462}\right)\) | \(e\left(\frac{6}{77}\right)\) | \(e\left(\frac{25}{77}\right)\) | \(e\left(\frac{307}{462}\right)\) | \(e\left(\frac{31}{231}\right)\) | \(e\left(\frac{401}{462}\right)\) | \(e\left(\frac{101}{231}\right)\) |
| \(\chi_{6003}(1211,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{231}\right)\) | \(e\left(\frac{142}{231}\right)\) | \(e\left(\frac{85}{231}\right)\) | \(e\left(\frac{73}{462}\right)\) | \(e\left(\frac{71}{77}\right)\) | \(e\left(\frac{52}{77}\right)\) | \(e\left(\frac{1}{462}\right)\) | \(e\left(\frac{46}{231}\right)\) | \(e\left(\frac{215}{462}\right)\) | \(e\left(\frac{53}{231}\right)\) |
| \(\chi_{6003}(1253,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{46}{231}\right)\) | \(e\left(\frac{92}{231}\right)\) | \(e\left(\frac{221}{231}\right)\) | \(e\left(\frac{5}{462}\right)\) | \(e\left(\frac{46}{77}\right)\) | \(e\left(\frac{12}{77}\right)\) | \(e\left(\frac{95}{462}\right)\) | \(e\left(\frac{212}{231}\right)\) | \(e\left(\frac{97}{462}\right)\) | \(e\left(\frac{184}{231}\right)\) |
| \(\chi_{6003}(1256,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{231}\right)\) | \(e\left(\frac{46}{231}\right)\) | \(e\left(\frac{226}{231}\right)\) | \(e\left(\frac{349}{462}\right)\) | \(e\left(\frac{23}{77}\right)\) | \(e\left(\frac{6}{77}\right)\) | \(e\left(\frac{163}{462}\right)\) | \(e\left(\frac{106}{231}\right)\) | \(e\left(\frac{395}{462}\right)\) | \(e\left(\frac{92}{231}\right)\) |
| \(\chi_{6003}(1280,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{181}{231}\right)\) | \(e\left(\frac{131}{231}\right)\) | \(e\left(\frac{41}{231}\right)\) | \(e\left(\frac{95}{462}\right)\) | \(e\left(\frac{27}{77}\right)\) | \(e\left(\frac{74}{77}\right)\) | \(e\left(\frac{419}{462}\right)\) | \(e\left(\frac{101}{231}\right)\) | \(e\left(\frac{457}{462}\right)\) | \(e\left(\frac{31}{231}\right)\) |
| \(\chi_{6003}(1298,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{85}{231}\right)\) | \(e\left(\frac{170}{231}\right)\) | \(e\left(\frac{92}{231}\right)\) | \(e\left(\frac{185}{462}\right)\) | \(e\left(\frac{8}{77}\right)\) | \(e\left(\frac{59}{77}\right)\) | \(e\left(\frac{281}{462}\right)\) | \(e\left(\frac{221}{231}\right)\) | \(e\left(\frac{355}{462}\right)\) | \(e\left(\frac{109}{231}\right)\) |
| \(\chi_{6003}(1397,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{136}{231}\right)\) | \(e\left(\frac{41}{231}\right)\) | \(e\left(\frac{101}{231}\right)\) | \(e\left(\frac{65}{462}\right)\) | \(e\left(\frac{59}{77}\right)\) | \(e\left(\frac{2}{77}\right)\) | \(e\left(\frac{311}{462}\right)\) | \(e\left(\frac{215}{231}\right)\) | \(e\left(\frac{337}{462}\right)\) | \(e\left(\frac{82}{231}\right)\) |
| \(\chi_{6003}(1454,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{230}{231}\right)\) | \(e\left(\frac{229}{231}\right)\) | \(e\left(\frac{181}{231}\right)\) | \(e\left(\frac{25}{462}\right)\) | \(e\left(\frac{76}{77}\right)\) | \(e\left(\frac{60}{77}\right)\) | \(e\left(\frac{13}{462}\right)\) | \(e\left(\frac{136}{231}\right)\) | \(e\left(\frac{23}{462}\right)\) | \(e\left(\frac{227}{231}\right)\) |
| \(\chi_{6003}(1463,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{231}\right)\) | \(e\left(\frac{178}{231}\right)\) | \(e\left(\frac{61}{231}\right)\) | \(e\left(\frac{85}{462}\right)\) | \(e\left(\frac{12}{77}\right)\) | \(e\left(\frac{50}{77}\right)\) | \(e\left(\frac{229}{462}\right)\) | \(e\left(\frac{139}{231}\right)\) | \(e\left(\frac{263}{462}\right)\) | \(e\left(\frac{125}{231}\right)\) |
| \(\chi_{6003}(1514,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{172}{231}\right)\) | \(e\left(\frac{113}{231}\right)\) | \(e\left(\frac{53}{231}\right)\) | \(e\left(\frac{89}{462}\right)\) | \(e\left(\frac{18}{77}\right)\) | \(e\left(\frac{75}{77}\right)\) | \(e\left(\frac{305}{462}\right)\) | \(e\left(\frac{170}{231}\right)\) | \(e\left(\frac{433}{462}\right)\) | \(e\left(\frac{226}{231}\right)\) |