from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3381, base_ring=CyclotomicField(462))
M = H._module
chi = DirichletCharacter(H, M([231,319,21]))
chi.galois_orbit()
[g,chi] = znchar(Mod(5,3381))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3381\) | |
| Conductor: | \(3381\) | 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 | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3381}(5,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{251}{462}\right)\) | \(e\left(\frac{20}{231}\right)\) | \(e\left(\frac{263}{462}\right)\) | \(e\left(\frac{97}{154}\right)\) | \(e\left(\frac{26}{231}\right)\) | \(e\left(\frac{122}{231}\right)\) | \(e\left(\frac{65}{154}\right)\) | \(e\left(\frac{40}{231}\right)\) | \(e\left(\frac{37}{462}\right)\) | \(e\left(\frac{28}{33}\right)\) |
| \(\chi_{3381}(17,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{283}{462}\right)\) | \(e\left(\frac{52}{231}\right)\) | \(e\left(\frac{37}{462}\right)\) | \(e\left(\frac{129}{154}\right)\) | \(e\left(\frac{160}{231}\right)\) | \(e\left(\frac{40}{231}\right)\) | \(e\left(\frac{15}{154}\right)\) | \(e\left(\frac{104}{231}\right)\) | \(e\left(\frac{281}{462}\right)\) | \(e\left(\frac{20}{33}\right)\) |
| \(\chi_{3381}(38,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{373}{462}\right)\) | \(e\left(\frac{142}{231}\right)\) | \(e\left(\frac{181}{462}\right)\) | \(e\left(\frac{65}{154}\right)\) | \(e\left(\frac{46}{231}\right)\) | \(e\left(\frac{127}{231}\right)\) | \(e\left(\frac{115}{154}\right)\) | \(e\left(\frac{53}{231}\right)\) | \(e\left(\frac{101}{462}\right)\) | \(e\left(\frac{14}{33}\right)\) |
| \(\chi_{3381}(89,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{462}\right)\) | \(e\left(\frac{89}{231}\right)\) | \(e\left(\frac{281}{462}\right)\) | \(e\left(\frac{89}{154}\right)\) | \(e\left(\frac{185}{231}\right)\) | \(e\left(\frac{104}{231}\right)\) | \(e\left(\frac{39}{154}\right)\) | \(e\left(\frac{178}{231}\right)\) | \(e\left(\frac{361}{462}\right)\) | \(e\left(\frac{19}{33}\right)\) |
| \(\chi_{3381}(122,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{462}\right)\) | \(e\left(\frac{61}{231}\right)\) | \(e\left(\frac{421}{462}\right)\) | \(e\left(\frac{61}{154}\right)\) | \(e\left(\frac{10}{231}\right)\) | \(e\left(\frac{118}{231}\right)\) | \(e\left(\frac{25}{154}\right)\) | \(e\left(\frac{122}{231}\right)\) | \(e\left(\frac{263}{462}\right)\) | \(e\left(\frac{26}{33}\right)\) |
| \(\chi_{3381}(143,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{361}{462}\right)\) | \(e\left(\frac{130}{231}\right)\) | \(e\left(\frac{439}{462}\right)\) | \(e\left(\frac{53}{154}\right)\) | \(e\left(\frac{169}{231}\right)\) | \(e\left(\frac{100}{231}\right)\) | \(e\left(\frac{153}{154}\right)\) | \(e\left(\frac{29}{231}\right)\) | \(e\left(\frac{125}{462}\right)\) | \(e\left(\frac{17}{33}\right)\) |
| \(\chi_{3381}(152,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{167}{462}\right)\) | \(e\left(\frac{167}{231}\right)\) | \(e\left(\frac{221}{462}\right)\) | \(e\left(\frac{13}{154}\right)\) | \(e\left(\frac{194}{231}\right)\) | \(e\left(\frac{164}{231}\right)\) | \(e\left(\frac{23}{154}\right)\) | \(e\left(\frac{103}{231}\right)\) | \(e\left(\frac{205}{462}\right)\) | \(e\left(\frac{16}{33}\right)\) |
| \(\chi_{3381}(194,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{401}{462}\right)\) | \(e\left(\frac{170}{231}\right)\) | \(e\left(\frac{41}{462}\right)\) | \(e\left(\frac{93}{154}\right)\) | \(e\left(\frac{221}{231}\right)\) | \(e\left(\frac{113}{231}\right)\) | \(e\left(\frac{129}{154}\right)\) | \(e\left(\frac{109}{231}\right)\) | \(e\left(\frac{199}{462}\right)\) | \(e\left(\frac{7}{33}\right)\) |
| \(\chi_{3381}(290,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{277}{462}\right)\) | \(e\left(\frac{46}{231}\right)\) | \(e\left(\frac{397}{462}\right)\) | \(e\left(\frac{123}{154}\right)\) | \(e\left(\frac{106}{231}\right)\) | \(e\left(\frac{142}{231}\right)\) | \(e\left(\frac{111}{154}\right)\) | \(e\left(\frac{92}{231}\right)\) | \(e\left(\frac{293}{462}\right)\) | \(e\left(\frac{5}{33}\right)\) |
| \(\chi_{3381}(320,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{95}{462}\right)\) | \(e\left(\frac{95}{231}\right)\) | \(e\left(\frac{383}{462}\right)\) | \(e\left(\frac{95}{154}\right)\) | \(e\left(\frac{8}{231}\right)\) | \(e\left(\frac{2}{231}\right)\) | \(e\left(\frac{97}{154}\right)\) | \(e\left(\frac{190}{231}\right)\) | \(e\left(\frac{349}{462}\right)\) | \(e\left(\frac{1}{33}\right)\) |
| \(\chi_{3381}(332,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{247}{462}\right)\) | \(e\left(\frac{16}{231}\right)\) | \(e\left(\frac{349}{462}\right)\) | \(e\left(\frac{93}{154}\right)\) | \(e\left(\frac{67}{231}\right)\) | \(e\left(\frac{190}{231}\right)\) | \(e\left(\frac{129}{154}\right)\) | \(e\left(\frac{32}{231}\right)\) | \(e\left(\frac{353}{462}\right)\) | \(e\left(\frac{29}{33}\right)\) |
| \(\chi_{3381}(341,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{443}{462}\right)\) | \(e\left(\frac{212}{231}\right)\) | \(e\left(\frac{293}{462}\right)\) | \(e\left(\frac{135}{154}\right)\) | \(e\left(\frac{137}{231}\right)\) | \(e\left(\frac{92}{231}\right)\) | \(e\left(\frac{73}{154}\right)\) | \(e\left(\frac{193}{231}\right)\) | \(e\left(\frac{115}{462}\right)\) | \(e\left(\frac{13}{33}\right)\) |
| \(\chi_{3381}(383,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{462}\right)\) | \(e\left(\frac{131}{231}\right)\) | \(e\left(\frac{71}{462}\right)\) | \(e\left(\frac{131}{154}\right)\) | \(e\left(\frac{101}{231}\right)\) | \(e\left(\frac{83}{231}\right)\) | \(e\left(\frac{137}{154}\right)\) | \(e\left(\frac{31}{231}\right)\) | \(e\left(\frac{277}{462}\right)\) | \(e\left(\frac{25}{33}\right)\) |
| \(\chi_{3381}(425,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{323}{462}\right)\) | \(e\left(\frac{92}{231}\right)\) | \(e\left(\frac{101}{462}\right)\) | \(e\left(\frac{15}{154}\right)\) | \(e\left(\frac{212}{231}\right)\) | \(e\left(\frac{53}{231}\right)\) | \(e\left(\frac{145}{154}\right)\) | \(e\left(\frac{184}{231}\right)\) | \(e\left(\frac{355}{462}\right)\) | \(e\left(\frac{10}{33}\right)\) |
| \(\chi_{3381}(458,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{462}\right)\) | \(e\left(\frac{73}{231}\right)\) | \(e\left(\frac{163}{462}\right)\) | \(e\left(\frac{73}{154}\right)\) | \(e\left(\frac{118}{231}\right)\) | \(e\left(\frac{145}{231}\right)\) | \(e\left(\frac{141}{154}\right)\) | \(e\left(\frac{146}{231}\right)\) | \(e\left(\frac{239}{462}\right)\) | \(e\left(\frac{23}{33}\right)\) |
| \(\chi_{3381}(467,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{347}{462}\right)\) | \(e\left(\frac{116}{231}\right)\) | \(e\left(\frac{47}{462}\right)\) | \(e\left(\frac{39}{154}\right)\) | \(e\left(\frac{197}{231}\right)\) | \(e\left(\frac{107}{231}\right)\) | \(e\left(\frac{69}{154}\right)\) | \(e\left(\frac{1}{231}\right)\) | \(e\left(\frac{307}{462}\right)\) | \(e\left(\frac{4}{33}\right)\) |
| \(\chi_{3381}(479,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{289}{462}\right)\) | \(e\left(\frac{58}{231}\right)\) | \(e\left(\frac{139}{462}\right)\) | \(e\left(\frac{135}{154}\right)\) | \(e\left(\frac{214}{231}\right)\) | \(e\left(\frac{169}{231}\right)\) | \(e\left(\frac{73}{154}\right)\) | \(e\left(\frac{116}{231}\right)\) | \(e\left(\frac{269}{462}\right)\) | \(e\left(\frac{2}{33}\right)\) |
| \(\chi_{3381}(488,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{317}{462}\right)\) | \(e\left(\frac{86}{231}\right)\) | \(e\left(\frac{461}{462}\right)\) | \(e\left(\frac{9}{154}\right)\) | \(e\left(\frac{158}{231}\right)\) | \(e\left(\frac{155}{231}\right)\) | \(e\left(\frac{87}{154}\right)\) | \(e\left(\frac{172}{231}\right)\) | \(e\left(\frac{367}{462}\right)\) | \(e\left(\frac{28}{33}\right)\) |
| \(\chi_{3381}(500,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{85}{462}\right)\) | \(e\left(\frac{85}{231}\right)\) | \(e\left(\frac{367}{462}\right)\) | \(e\left(\frac{85}{154}\right)\) | \(e\left(\frac{226}{231}\right)\) | \(e\left(\frac{172}{231}\right)\) | \(e\left(\frac{103}{154}\right)\) | \(e\left(\frac{170}{231}\right)\) | \(e\left(\frac{215}{462}\right)\) | \(e\left(\frac{20}{33}\right)\) |
| \(\chi_{3381}(563,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{103}{462}\right)\) | \(e\left(\frac{103}{231}\right)\) | \(e\left(\frac{211}{462}\right)\) | \(e\left(\frac{103}{154}\right)\) | \(e\left(\frac{157}{231}\right)\) | \(e\left(\frac{97}{231}\right)\) | \(e\left(\frac{123}{154}\right)\) | \(e\left(\frac{206}{231}\right)\) | \(e\left(\frac{179}{462}\right)\) | \(e\left(\frac{32}{33}\right)\) |
| \(\chi_{3381}(572,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{155}{462}\right)\) | \(e\left(\frac{155}{231}\right)\) | \(e\left(\frac{17}{462}\right)\) | \(e\left(\frac{1}{154}\right)\) | \(e\left(\frac{86}{231}\right)\) | \(e\left(\frac{137}{231}\right)\) | \(e\left(\frac{61}{154}\right)\) | \(e\left(\frac{79}{231}\right)\) | \(e\left(\frac{229}{462}\right)\) | \(e\left(\frac{19}{33}\right)\) |
| \(\chi_{3381}(605,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{325}{462}\right)\) | \(e\left(\frac{94}{231}\right)\) | \(e\left(\frac{289}{462}\right)\) | \(e\left(\frac{17}{154}\right)\) | \(e\left(\frac{76}{231}\right)\) | \(e\left(\frac{19}{231}\right)\) | \(e\left(\frac{113}{154}\right)\) | \(e\left(\frac{188}{231}\right)\) | \(e\left(\frac{197}{462}\right)\) | \(e\left(\frac{26}{33}\right)\) |
| \(\chi_{3381}(626,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{163}{462}\right)\) | \(e\left(\frac{163}{231}\right)\) | \(e\left(\frac{307}{462}\right)\) | \(e\left(\frac{9}{154}\right)\) | \(e\left(\frac{4}{231}\right)\) | \(e\left(\frac{1}{231}\right)\) | \(e\left(\frac{87}{154}\right)\) | \(e\left(\frac{95}{231}\right)\) | \(e\left(\frac{59}{462}\right)\) | \(e\left(\frac{17}{33}\right)\) |
| \(\chi_{3381}(635,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{233}{462}\right)\) | \(e\left(\frac{2}{231}\right)\) | \(e\left(\frac{419}{462}\right)\) | \(e\left(\frac{79}{154}\right)\) | \(e\left(\frac{95}{231}\right)\) | \(e\left(\frac{197}{231}\right)\) | \(e\left(\frac{45}{154}\right)\) | \(e\left(\frac{4}{231}\right)\) | \(e\left(\frac{73}{462}\right)\) | \(e\left(\frac{16}{33}\right)\) |
| \(\chi_{3381}(677,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{462}\right)\) | \(e\left(\frac{5}{231}\right)\) | \(e\left(\frac{239}{462}\right)\) | \(e\left(\frac{5}{154}\right)\) | \(e\left(\frac{122}{231}\right)\) | \(e\left(\frac{146}{231}\right)\) | \(e\left(\frac{151}{154}\right)\) | \(e\left(\frac{10}{231}\right)\) | \(e\left(\frac{67}{462}\right)\) | \(e\left(\frac{7}{33}\right)\) |
| \(\chi_{3381}(710,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{397}{462}\right)\) | \(e\left(\frac{166}{231}\right)\) | \(e\left(\frac{127}{462}\right)\) | \(e\left(\frac{89}{154}\right)\) | \(e\left(\frac{31}{231}\right)\) | \(e\left(\frac{181}{231}\right)\) | \(e\left(\frac{39}{154}\right)\) | \(e\left(\frac{101}{231}\right)\) | \(e\left(\frac{53}{462}\right)\) | \(e\left(\frac{8}{33}\right)\) |
| \(\chi_{3381}(773,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{462}\right)\) | \(e\left(\frac{79}{231}\right)\) | \(e\left(\frac{265}{462}\right)\) | \(e\left(\frac{79}{154}\right)\) | \(e\left(\frac{172}{231}\right)\) | \(e\left(\frac{43}{231}\right)\) | \(e\left(\frac{45}{154}\right)\) | \(e\left(\frac{158}{231}\right)\) | \(e\left(\frac{227}{462}\right)\) | \(e\left(\frac{5}{33}\right)\) |
| \(\chi_{3381}(824,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{462}\right)\) | \(e\left(\frac{47}{231}\right)\) | \(e\left(\frac{29}{462}\right)\) | \(e\left(\frac{47}{154}\right)\) | \(e\left(\frac{38}{231}\right)\) | \(e\left(\frac{125}{231}\right)\) | \(e\left(\frac{95}{154}\right)\) | \(e\left(\frac{94}{231}\right)\) | \(e\left(\frac{445}{462}\right)\) | \(e\left(\frac{13}{33}\right)\) |
| \(\chi_{3381}(845,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{437}{462}\right)\) | \(e\left(\frac{206}{231}\right)\) | \(e\left(\frac{191}{462}\right)\) | \(e\left(\frac{129}{154}\right)\) | \(e\left(\frac{83}{231}\right)\) | \(e\left(\frac{194}{231}\right)\) | \(e\left(\frac{15}{154}\right)\) | \(e\left(\frac{181}{231}\right)\) | \(e\left(\frac{127}{462}\right)\) | \(e\left(\frac{31}{33}\right)\) |
| \(\chi_{3381}(866,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{197}{462}\right)\) | \(e\left(\frac{197}{231}\right)\) | \(e\left(\frac{269}{462}\right)\) | \(e\left(\frac{43}{154}\right)\) | \(e\left(\frac{2}{231}\right)\) | \(e\left(\frac{116}{231}\right)\) | \(e\left(\frac{5}{154}\right)\) | \(e\left(\frac{163}{231}\right)\) | \(e\left(\frac{145}{462}\right)\) | \(e\left(\frac{25}{33}\right)\) |
| \(\chi_{3381}(908,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{389}{462}\right)\) | \(e\left(\frac{158}{231}\right)\) | \(e\left(\frac{299}{462}\right)\) | \(e\left(\frac{81}{154}\right)\) | \(e\left(\frac{113}{231}\right)\) | \(e\left(\frac{86}{231}\right)\) | \(e\left(\frac{13}{154}\right)\) | \(e\left(\frac{85}{231}\right)\) | \(e\left(\frac{223}{462}\right)\) | \(e\left(\frac{10}{33}\right)\) |