from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6003, base_ring=CyclotomicField(462))
M = H._module
chi = DirichletCharacter(H, M([308,168,66]))
chi.galois_orbit()
[g,chi] = znchar(Mod(16,6003))
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: | \(231\) | 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 231 polynomial (not computed) |
First 31 of 120 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{6003}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{124}{231}\right)\) | \(e\left(\frac{17}{231}\right)\) | \(e\left(\frac{194}{231}\right)\) | \(e\left(\frac{67}{231}\right)\) | \(e\left(\frac{47}{77}\right)\) | \(e\left(\frac{29}{77}\right)\) | \(e\left(\frac{118}{231}\right)\) | \(e\left(\frac{230}{231}\right)\) | \(e\left(\frac{191}{231}\right)\) | \(e\left(\frac{34}{231}\right)\) |
| \(\chi_{6003}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{231}\right)\) | \(e\left(\frac{194}{231}\right)\) | \(e\left(\frac{230}{231}\right)\) | \(e\left(\frac{58}{231}\right)\) | \(e\left(\frac{20}{77}\right)\) | \(e\left(\frac{32}{77}\right)\) | \(e\left(\frac{178}{231}\right)\) | \(e\left(\frac{206}{231}\right)\) | \(e\left(\frac{155}{231}\right)\) | \(e\left(\frac{157}{231}\right)\) |
| \(\chi_{6003}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{149}{231}\right)\) | \(e\left(\frac{67}{231}\right)\) | \(e\left(\frac{58}{231}\right)\) | \(e\left(\frac{101}{231}\right)\) | \(e\left(\frac{72}{77}\right)\) | \(e\left(\frac{69}{77}\right)\) | \(e\left(\frac{71}{231}\right)\) | \(e\left(\frac{64}{231}\right)\) | \(e\left(\frac{19}{231}\right)\) | \(e\left(\frac{134}{231}\right)\) |
| \(\chi_{6003}(52,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{231}\right)\) | \(e\left(\frac{8}{231}\right)\) | \(e\left(\frac{200}{231}\right)\) | \(e\left(\frac{181}{231}\right)\) | \(e\left(\frac{4}{77}\right)\) | \(e\left(\frac{68}{77}\right)\) | \(e\left(\frac{205}{231}\right)\) | \(e\left(\frac{149}{231}\right)\) | \(e\left(\frac{185}{231}\right)\) | \(e\left(\frac{16}{231}\right)\) |
| \(\chi_{6003}(94,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{218}{231}\right)\) | \(e\left(\frac{205}{231}\right)\) | \(e\left(\frac{43}{231}\right)\) | \(e\left(\frac{47}{231}\right)\) | \(e\left(\frac{64}{77}\right)\) | \(e\left(\frac{10}{77}\right)\) | \(e\left(\frac{200}{231}\right)\) | \(e\left(\frac{151}{231}\right)\) | \(e\left(\frac{34}{231}\right)\) | \(e\left(\frac{179}{231}\right)\) |
| \(\chi_{6003}(169,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{115}{231}\right)\) | \(e\left(\frac{230}{231}\right)\) | \(e\left(\frac{206}{231}\right)\) | \(e\left(\frac{64}{231}\right)\) | \(e\left(\frac{38}{77}\right)\) | \(e\left(\frac{30}{77}\right)\) | \(e\left(\frac{61}{231}\right)\) | \(e\left(\frac{68}{231}\right)\) | \(e\left(\frac{179}{231}\right)\) | \(e\left(\frac{229}{231}\right)\) |
| \(\chi_{6003}(223,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{58}{231}\right)\) | \(e\left(\frac{116}{231}\right)\) | \(e\left(\frac{128}{231}\right)\) | \(e\left(\frac{199}{231}\right)\) | \(e\left(\frac{58}{77}\right)\) | \(e\left(\frac{62}{77}\right)\) | \(e\left(\frac{85}{231}\right)\) | \(e\left(\frac{197}{231}\right)\) | \(e\left(\frac{26}{231}\right)\) | \(e\left(\frac{1}{231}\right)\) |
| \(\chi_{6003}(256,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{231}\right)\) | \(e\left(\frac{34}{231}\right)\) | \(e\left(\frac{157}{231}\right)\) | \(e\left(\frac{134}{231}\right)\) | \(e\left(\frac{17}{77}\right)\) | \(e\left(\frac{58}{77}\right)\) | \(e\left(\frac{5}{231}\right)\) | \(e\left(\frac{229}{231}\right)\) | \(e\left(\frac{151}{231}\right)\) | \(e\left(\frac{68}{231}\right)\) |
| \(\chi_{6003}(400,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{221}{231}\right)\) | \(e\left(\frac{211}{231}\right)\) | \(e\left(\frac{193}{231}\right)\) | \(e\left(\frac{125}{231}\right)\) | \(e\left(\frac{67}{77}\right)\) | \(e\left(\frac{61}{77}\right)\) | \(e\left(\frac{65}{231}\right)\) | \(e\left(\frac{205}{231}\right)\) | \(e\left(\frac{115}{231}\right)\) | \(e\left(\frac{191}{231}\right)\) |
| \(\chi_{6003}(430,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{157}{231}\right)\) | \(e\left(\frac{83}{231}\right)\) | \(e\left(\frac{227}{231}\right)\) | \(e\left(\frac{1}{231}\right)\) | \(e\left(\frac{3}{77}\right)\) | \(e\left(\frac{51}{77}\right)\) | \(e\left(\frac{19}{231}\right)\) | \(e\left(\frac{131}{231}\right)\) | \(e\left(\frac{158}{231}\right)\) | \(e\left(\frac{166}{231}\right)\) |
| \(\chi_{6003}(538,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{166}{231}\right)\) | \(e\left(\frac{101}{231}\right)\) | \(e\left(\frac{215}{231}\right)\) | \(e\left(\frac{4}{231}\right)\) | \(e\left(\frac{12}{77}\right)\) | \(e\left(\frac{50}{77}\right)\) | \(e\left(\frac{76}{231}\right)\) | \(e\left(\frac{62}{231}\right)\) | \(e\left(\frac{170}{231}\right)\) | \(e\left(\frac{202}{231}\right)\) |
| \(\chi_{6003}(547,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{76}{231}\right)\) | \(e\left(\frac{152}{231}\right)\) | \(e\left(\frac{104}{231}\right)\) | \(e\left(\frac{205}{231}\right)\) | \(e\left(\frac{76}{77}\right)\) | \(e\left(\frac{60}{77}\right)\) | \(e\left(\frac{199}{231}\right)\) | \(e\left(\frac{59}{231}\right)\) | \(e\left(\frac{50}{231}\right)\) | \(e\left(\frac{73}{231}\right)\) |
| \(\chi_{6003}(616,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{197}{231}\right)\) | \(e\left(\frac{163}{231}\right)\) | \(e\left(\frac{148}{231}\right)\) | \(e\left(\frac{194}{231}\right)\) | \(e\left(\frac{43}{77}\right)\) | \(e\left(\frac{38}{77}\right)\) | \(e\left(\frac{221}{231}\right)\) | \(e\left(\frac{4}{231}\right)\) | \(e\left(\frac{160}{231}\right)\) | \(e\left(\frac{95}{231}\right)\) |
| \(\chi_{6003}(625,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{194}{231}\right)\) | \(e\left(\frac{157}{231}\right)\) | \(e\left(\frac{229}{231}\right)\) | \(e\left(\frac{116}{231}\right)\) | \(e\left(\frac{40}{77}\right)\) | \(e\left(\frac{64}{77}\right)\) | \(e\left(\frac{125}{231}\right)\) | \(e\left(\frac{181}{231}\right)\) | \(e\left(\frac{79}{231}\right)\) | \(e\left(\frac{83}{231}\right)\) |
| \(\chi_{6003}(634,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{231}\right)\) | \(e\left(\frac{82}{231}\right)\) | \(e\left(\frac{202}{231}\right)\) | \(e\left(\frac{65}{231}\right)\) | \(e\left(\frac{41}{77}\right)\) | \(e\left(\frac{4}{77}\right)\) | \(e\left(\frac{80}{231}\right)\) | \(e\left(\frac{199}{231}\right)\) | \(e\left(\frac{106}{231}\right)\) | \(e\left(\frac{164}{231}\right)\) |
| \(\chi_{6003}(745,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{100}{231}\right)\) | \(e\left(\frac{200}{231}\right)\) | \(e\left(\frac{149}{231}\right)\) | \(e\left(\frac{136}{231}\right)\) | \(e\left(\frac{23}{77}\right)\) | \(e\left(\frac{6}{77}\right)\) | \(e\left(\frac{43}{231}\right)\) | \(e\left(\frac{29}{231}\right)\) | \(e\left(\frac{5}{231}\right)\) | \(e\left(\frac{169}{231}\right)\) |
| \(\chi_{6003}(790,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{148}{231}\right)\) | \(e\left(\frac{65}{231}\right)\) | \(e\left(\frac{8}{231}\right)\) | \(e\left(\frac{229}{231}\right)\) | \(e\left(\frac{71}{77}\right)\) | \(e\left(\frac{52}{77}\right)\) | \(e\left(\frac{193}{231}\right)\) | \(e\left(\frac{200}{231}\right)\) | \(e\left(\frac{146}{231}\right)\) | \(e\left(\frac{130}{231}\right)\) |
| \(\chi_{6003}(808,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{160}{231}\right)\) | \(e\left(\frac{89}{231}\right)\) | \(e\left(\frac{146}{231}\right)\) | \(e\left(\frac{79}{231}\right)\) | \(e\left(\frac{6}{77}\right)\) | \(e\left(\frac{25}{77}\right)\) | \(e\left(\frac{115}{231}\right)\) | \(e\left(\frac{185}{231}\right)\) | \(e\left(\frac{8}{231}\right)\) | \(e\left(\frac{178}{231}\right)\) |
| \(\chi_{6003}(832,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{128}{231}\right)\) | \(e\left(\frac{25}{231}\right)\) | \(e\left(\frac{163}{231}\right)\) | \(e\left(\frac{17}{231}\right)\) | \(e\left(\frac{51}{77}\right)\) | \(e\left(\frac{20}{77}\right)\) | \(e\left(\frac{92}{231}\right)\) | \(e\left(\frac{148}{231}\right)\) | \(e\left(\frac{145}{231}\right)\) | \(e\left(\frac{50}{231}\right)\) |
| \(\chi_{6003}(877,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{50}{231}\right)\) | \(e\left(\frac{100}{231}\right)\) | \(e\left(\frac{190}{231}\right)\) | \(e\left(\frac{68}{231}\right)\) | \(e\left(\frac{50}{77}\right)\) | \(e\left(\frac{3}{77}\right)\) | \(e\left(\frac{137}{231}\right)\) | \(e\left(\frac{130}{231}\right)\) | \(e\left(\frac{118}{231}\right)\) | \(e\left(\frac{200}{231}\right)\) |
| \(\chi_{6003}(886,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{68}{231}\right)\) | \(e\left(\frac{136}{231}\right)\) | \(e\left(\frac{166}{231}\right)\) | \(e\left(\frac{74}{231}\right)\) | \(e\left(\frac{68}{77}\right)\) | \(e\left(\frac{1}{77}\right)\) | \(e\left(\frac{20}{231}\right)\) | \(e\left(\frac{223}{231}\right)\) | \(e\left(\frac{142}{231}\right)\) | \(e\left(\frac{41}{231}\right)\) |
| \(\chi_{6003}(922,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{231}\right)\) | \(e\left(\frac{106}{231}\right)\) | \(e\left(\frac{109}{231}\right)\) | \(e\left(\frac{146}{231}\right)\) | \(e\left(\frac{53}{77}\right)\) | \(e\left(\frac{54}{77}\right)\) | \(e\left(\frac{2}{231}\right)\) | \(e\left(\frac{184}{231}\right)\) | \(e\left(\frac{199}{231}\right)\) | \(e\left(\frac{212}{231}\right)\) |
| \(\chi_{6003}(952,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{199}{231}\right)\) | \(e\left(\frac{167}{231}\right)\) | \(e\left(\frac{17}{231}\right)\) | \(e\left(\frac{169}{231}\right)\) | \(e\left(\frac{45}{77}\right)\) | \(e\left(\frac{72}{77}\right)\) | \(e\left(\frac{208}{231}\right)\) | \(e\left(\frac{194}{231}\right)\) | \(e\left(\frac{137}{231}\right)\) | \(e\left(\frac{103}{231}\right)\) |
| \(\chi_{6003}(1039,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{227}{231}\right)\) | \(e\left(\frac{223}{231}\right)\) | \(e\left(\frac{31}{231}\right)\) | \(e\left(\frac{50}{231}\right)\) | \(e\left(\frac{73}{77}\right)\) | \(e\left(\frac{9}{77}\right)\) | \(e\left(\frac{26}{231}\right)\) | \(e\left(\frac{82}{231}\right)\) | \(e\left(\frac{46}{231}\right)\) | \(e\left(\frac{215}{231}\right)\) |
| \(\chi_{6003}(1051,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{190}{231}\right)\) | \(e\left(\frac{149}{231}\right)\) | \(e\left(\frac{29}{231}\right)\) | \(e\left(\frac{166}{231}\right)\) | \(e\left(\frac{36}{77}\right)\) | \(e\left(\frac{73}{77}\right)\) | \(e\left(\frac{151}{231}\right)\) | \(e\left(\frac{32}{231}\right)\) | \(e\left(\frac{125}{231}\right)\) | \(e\left(\frac{67}{231}\right)\) |
| \(\chi_{6003}(1060,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{229}{231}\right)\) | \(e\left(\frac{227}{231}\right)\) | \(e\left(\frac{131}{231}\right)\) | \(e\left(\frac{25}{231}\right)\) | \(e\left(\frac{75}{77}\right)\) | \(e\left(\frac{43}{77}\right)\) | \(e\left(\frac{13}{231}\right)\) | \(e\left(\frac{41}{231}\right)\) | \(e\left(\frac{23}{231}\right)\) | \(e\left(\frac{223}{231}\right)\) |
| \(\chi_{6003}(1093,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{231}\right)\) | \(e\left(\frac{4}{231}\right)\) | \(e\left(\frac{100}{231}\right)\) | \(e\left(\frac{206}{231}\right)\) | \(e\left(\frac{2}{77}\right)\) | \(e\left(\frac{34}{77}\right)\) | \(e\left(\frac{218}{231}\right)\) | \(e\left(\frac{190}{231}\right)\) | \(e\left(\frac{208}{231}\right)\) | \(e\left(\frac{8}{231}\right)\) |
| \(\chi_{6003}(1156,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{125}{231}\right)\) | \(e\left(\frac{19}{231}\right)\) | \(e\left(\frac{13}{231}\right)\) | \(e\left(\frac{170}{231}\right)\) | \(e\left(\frac{48}{77}\right)\) | \(e\left(\frac{46}{77}\right)\) | \(e\left(\frac{227}{231}\right)\) | \(e\left(\frac{94}{231}\right)\) | \(e\left(\frac{64}{231}\right)\) | \(e\left(\frac{38}{231}\right)\) |
| \(\chi_{6003}(1267,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{163}{231}\right)\) | \(e\left(\frac{95}{231}\right)\) | \(e\left(\frac{65}{231}\right)\) | \(e\left(\frac{157}{231}\right)\) | \(e\left(\frac{9}{77}\right)\) | \(e\left(\frac{76}{77}\right)\) | \(e\left(\frac{211}{231}\right)\) | \(e\left(\frac{8}{231}\right)\) | \(e\left(\frac{89}{231}\right)\) | \(e\left(\frac{190}{231}\right)\) |
| \(\chi_{6003}(1300,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{231}\right)\) | \(e\left(\frac{202}{231}\right)\) | \(e\left(\frac{199}{231}\right)\) | \(e\left(\frac{8}{231}\right)\) | \(e\left(\frac{24}{77}\right)\) | \(e\left(\frac{23}{77}\right)\) | \(e\left(\frac{152}{231}\right)\) | \(e\left(\frac{124}{231}\right)\) | \(e\left(\frac{109}{231}\right)\) | \(e\left(\frac{173}{231}\right)\) |
| \(\chi_{6003}(1444,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{231}\right)\) | \(e\left(\frac{64}{231}\right)\) | \(e\left(\frac{214}{231}\right)\) | \(e\left(\frac{62}{231}\right)\) | \(e\left(\frac{32}{77}\right)\) | \(e\left(\frac{5}{77}\right)\) | \(e\left(\frac{23}{231}\right)\) | \(e\left(\frac{37}{231}\right)\) | \(e\left(\frac{94}{231}\right)\) | \(e\left(\frac{128}{231}\right)\) |