from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3381, base_ring=CyclotomicField(462))
M = H._module
chi = DirichletCharacter(H, M([231,440,189]))
chi.galois_orbit()
[g,chi] = znchar(Mod(11,3381))
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: | 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 31 of 120 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3381}(11,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{462}\right)\) | \(e\left(\frac{37}{231}\right)\) | \(e\left(\frac{122}{231}\right)\) | \(e\left(\frac{37}{154}\right)\) | \(e\left(\frac{281}{462}\right)\) | \(e\left(\frac{64}{231}\right)\) | \(e\left(\frac{12}{77}\right)\) | \(e\left(\frac{74}{231}\right)\) | \(e\left(\frac{40}{231}\right)\) | \(e\left(\frac{31}{66}\right)\) |
| \(\chi_{3381}(44,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{293}{462}\right)\) | \(e\left(\frac{62}{231}\right)\) | \(e\left(\frac{142}{231}\right)\) | \(e\left(\frac{139}{154}\right)\) | \(e\left(\frac{115}{462}\right)\) | \(e\left(\frac{101}{231}\right)\) | \(e\left(\frac{43}{77}\right)\) | \(e\left(\frac{124}{231}\right)\) | \(e\left(\frac{92}{231}\right)\) | \(e\left(\frac{35}{66}\right)\) |
| \(\chi_{3381}(53,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{193}{462}\right)\) | \(e\left(\frac{193}{231}\right)\) | \(e\left(\frac{62}{231}\right)\) | \(e\left(\frac{39}{154}\right)\) | \(e\left(\frac{317}{462}\right)\) | \(e\left(\frac{184}{231}\right)\) | \(e\left(\frac{73}{77}\right)\) | \(e\left(\frac{155}{231}\right)\) | \(e\left(\frac{115}{231}\right)\) | \(e\left(\frac{19}{66}\right)\) |
| \(\chi_{3381}(65,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{462}\right)\) | \(e\left(\frac{113}{231}\right)\) | \(e\left(\frac{229}{231}\right)\) | \(e\left(\frac{113}{154}\right)\) | \(e\left(\frac{109}{462}\right)\) | \(e\left(\frac{158}{231}\right)\) | \(e\left(\frac{20}{77}\right)\) | \(e\left(\frac{226}{231}\right)\) | \(e\left(\frac{41}{231}\right)\) | \(e\left(\frac{59}{66}\right)\) |
| \(\chi_{3381}(74,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{229}{462}\right)\) | \(e\left(\frac{229}{231}\right)\) | \(e\left(\frac{137}{231}\right)\) | \(e\left(\frac{75}{154}\right)\) | \(e\left(\frac{41}{462}\right)\) | \(e\left(\frac{34}{231}\right)\) | \(e\left(\frac{16}{77}\right)\) | \(e\left(\frac{227}{231}\right)\) | \(e\left(\frac{79}{231}\right)\) | \(e\left(\frac{1}{66}\right)\) |
| \(\chi_{3381}(86,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{437}{462}\right)\) | \(e\left(\frac{206}{231}\right)\) | \(e\left(\frac{211}{231}\right)\) | \(e\left(\frac{129}{154}\right)\) | \(e\left(\frac{397}{462}\right)\) | \(e\left(\frac{194}{231}\right)\) | \(e\left(\frac{46}{77}\right)\) | \(e\left(\frac{181}{231}\right)\) | \(e\left(\frac{179}{231}\right)\) | \(e\left(\frac{29}{66}\right)\) |
| \(\chi_{3381}(107,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{131}{462}\right)\) | \(e\left(\frac{131}{231}\right)\) | \(e\left(\frac{151}{231}\right)\) | \(e\left(\frac{131}{154}\right)\) | \(e\left(\frac{433}{462}\right)\) | \(e\left(\frac{83}{231}\right)\) | \(e\left(\frac{30}{77}\right)\) | \(e\left(\frac{31}{231}\right)\) | \(e\left(\frac{23}{231}\right)\) | \(e\left(\frac{17}{66}\right)\) |
| \(\chi_{3381}(149,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{191}{462}\right)\) | \(e\left(\frac{191}{231}\right)\) | \(e\left(\frac{199}{231}\right)\) | \(e\left(\frac{37}{154}\right)\) | \(e\left(\frac{127}{462}\right)\) | \(e\left(\frac{218}{231}\right)\) | \(e\left(\frac{12}{77}\right)\) | \(e\left(\frac{151}{231}\right)\) | \(e\left(\frac{194}{231}\right)\) | \(e\left(\frac{53}{66}\right)\) |
| \(\chi_{3381}(158,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{331}{462}\right)\) | \(e\left(\frac{100}{231}\right)\) | \(e\left(\frac{80}{231}\right)\) | \(e\left(\frac{23}{154}\right)\) | \(e\left(\frac{29}{462}\right)\) | \(e\left(\frac{148}{231}\right)\) | \(e\left(\frac{47}{77}\right)\) | \(e\left(\frac{200}{231}\right)\) | \(e\left(\frac{208}{231}\right)\) | \(e\left(\frac{49}{66}\right)\) |
| \(\chi_{3381}(191,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{462}\right)\) | \(e\left(\frac{83}{231}\right)\) | \(e\left(\frac{205}{231}\right)\) | \(e\left(\frac{83}{154}\right)\) | \(e\left(\frac{31}{462}\right)\) | \(e\left(\frac{206}{231}\right)\) | \(e\left(\frac{29}{77}\right)\) | \(e\left(\frac{166}{231}\right)\) | \(e\left(\frac{71}{231}\right)\) | \(e\left(\frac{41}{66}\right)\) |
| \(\chi_{3381}(212,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{449}{462}\right)\) | \(e\left(\frac{218}{231}\right)\) | \(e\left(\frac{82}{231}\right)\) | \(e\left(\frac{141}{154}\right)\) | \(e\left(\frac{151}{462}\right)\) | \(e\left(\frac{221}{231}\right)\) | \(e\left(\frac{27}{77}\right)\) | \(e\left(\frac{205}{231}\right)\) | \(e\left(\frac{167}{231}\right)\) | \(e\left(\frac{23}{66}\right)\) |
| \(\chi_{3381}(221,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{145}{462}\right)\) | \(e\left(\frac{145}{231}\right)\) | \(e\left(\frac{116}{231}\right)\) | \(e\left(\frac{145}{154}\right)\) | \(e\left(\frac{377}{462}\right)\) | \(e\left(\frac{76}{231}\right)\) | \(e\left(\frac{72}{77}\right)\) | \(e\left(\frac{59}{231}\right)\) | \(e\left(\frac{163}{231}\right)\) | \(e\left(\frac{43}{66}\right)\) |
| \(\chi_{3381}(296,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{462}\right)\) | \(e\left(\frac{23}{231}\right)\) | \(e\left(\frac{157}{231}\right)\) | \(e\left(\frac{23}{154}\right)\) | \(e\left(\frac{337}{462}\right)\) | \(e\left(\frac{71}{231}\right)\) | \(e\left(\frac{47}{77}\right)\) | \(e\left(\frac{46}{231}\right)\) | \(e\left(\frac{131}{231}\right)\) | \(e\left(\frac{5}{66}\right)\) |
| \(\chi_{3381}(359,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{365}{462}\right)\) | \(e\left(\frac{134}{231}\right)\) | \(e\left(\frac{61}{231}\right)\) | \(e\left(\frac{57}{154}\right)\) | \(e\left(\frac{25}{462}\right)\) | \(e\left(\frac{32}{231}\right)\) | \(e\left(\frac{6}{77}\right)\) | \(e\left(\frac{37}{231}\right)\) | \(e\left(\frac{20}{231}\right)\) | \(e\left(\frac{65}{66}\right)\) |
| \(\chi_{3381}(389,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{139}{462}\right)\) | \(e\left(\frac{139}{231}\right)\) | \(e\left(\frac{65}{231}\right)\) | \(e\left(\frac{139}{154}\right)\) | \(e\left(\frac{269}{462}\right)\) | \(e\left(\frac{178}{231}\right)\) | \(e\left(\frac{43}{77}\right)\) | \(e\left(\frac{47}{231}\right)\) | \(e\left(\frac{169}{231}\right)\) | \(e\left(\frac{13}{66}\right)\) |
| \(\chi_{3381}(401,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{462}\right)\) | \(e\left(\frac{5}{231}\right)\) | \(e\left(\frac{4}{231}\right)\) | \(e\left(\frac{5}{154}\right)\) | \(e\left(\frac{13}{462}\right)\) | \(e\left(\frac{146}{231}\right)\) | \(e\left(\frac{37}{77}\right)\) | \(e\left(\frac{10}{231}\right)\) | \(e\left(\frac{149}{231}\right)\) | \(e\left(\frac{47}{66}\right)\) |
| \(\chi_{3381}(431,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{85}{462}\right)\) | \(e\left(\frac{85}{231}\right)\) | \(e\left(\frac{68}{231}\right)\) | \(e\left(\frac{85}{154}\right)\) | \(e\left(\frac{221}{462}\right)\) | \(e\left(\frac{172}{231}\right)\) | \(e\left(\frac{13}{77}\right)\) | \(e\left(\frac{170}{231}\right)\) | \(e\left(\frac{223}{231}\right)\) | \(e\left(\frac{7}{66}\right)\) |
| \(\chi_{3381}(452,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{373}{462}\right)\) | \(e\left(\frac{142}{231}\right)\) | \(e\left(\frac{206}{231}\right)\) | \(e\left(\frac{65}{154}\right)\) | \(e\left(\frac{323}{462}\right)\) | \(e\left(\frac{127}{231}\right)\) | \(e\left(\frac{19}{77}\right)\) | \(e\left(\frac{53}{231}\right)\) | \(e\left(\frac{166}{231}\right)\) | \(e\left(\frac{61}{66}\right)\) |
| \(\chi_{3381}(494,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{235}{462}\right)\) | \(e\left(\frac{4}{231}\right)\) | \(e\left(\frac{188}{231}\right)\) | \(e\left(\frac{81}{154}\right)\) | \(e\left(\frac{149}{462}\right)\) | \(e\left(\frac{163}{231}\right)\) | \(e\left(\frac{45}{77}\right)\) | \(e\left(\frac{8}{231}\right)\) | \(e\left(\frac{73}{231}\right)\) | \(e\left(\frac{31}{66}\right)\) |
| \(\chi_{3381}(527,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{227}{462}\right)\) | \(e\left(\frac{227}{231}\right)\) | \(e\left(\frac{43}{231}\right)\) | \(e\left(\frac{73}{154}\right)\) | \(e\left(\frac{313}{462}\right)\) | \(e\left(\frac{68}{231}\right)\) | \(e\left(\frac{32}{77}\right)\) | \(e\left(\frac{223}{231}\right)\) | \(e\left(\frac{158}{231}\right)\) | \(e\left(\frac{35}{66}\right)\) |
| \(\chi_{3381}(536,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{391}{462}\right)\) | \(e\left(\frac{160}{231}\right)\) | \(e\left(\frac{128}{231}\right)\) | \(e\left(\frac{83}{154}\right)\) | \(e\left(\frac{185}{462}\right)\) | \(e\left(\frac{52}{231}\right)\) | \(e\left(\frac{29}{77}\right)\) | \(e\left(\frac{89}{231}\right)\) | \(e\left(\frac{148}{231}\right)\) | \(e\left(\frac{19}{66}\right)\) |
| \(\chi_{3381}(548,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{462}\right)\) | \(e\left(\frac{47}{231}\right)\) | \(e\left(\frac{130}{231}\right)\) | \(e\left(\frac{47}{154}\right)\) | \(e\left(\frac{307}{462}\right)\) | \(e\left(\frac{125}{231}\right)\) | \(e\left(\frac{9}{77}\right)\) | \(e\left(\frac{94}{231}\right)\) | \(e\left(\frac{107}{231}\right)\) | \(e\left(\frac{59}{66}\right)\) |
| \(\chi_{3381}(590,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{65}{462}\right)\) | \(e\left(\frac{65}{231}\right)\) | \(e\left(\frac{52}{231}\right)\) | \(e\left(\frac{65}{154}\right)\) | \(e\left(\frac{169}{462}\right)\) | \(e\left(\frac{50}{231}\right)\) | \(e\left(\frac{19}{77}\right)\) | \(e\left(\frac{130}{231}\right)\) | \(e\left(\frac{89}{231}\right)\) | \(e\left(\frac{17}{66}\right)\) |
| \(\chi_{3381}(632,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{125}{462}\right)\) | \(e\left(\frac{125}{231}\right)\) | \(e\left(\frac{100}{231}\right)\) | \(e\left(\frac{125}{154}\right)\) | \(e\left(\frac{325}{462}\right)\) | \(e\left(\frac{185}{231}\right)\) | \(e\left(\frac{1}{77}\right)\) | \(e\left(\frac{19}{231}\right)\) | \(e\left(\frac{29}{231}\right)\) | \(e\left(\frac{53}{66}\right)\) |
| \(\chi_{3381}(641,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{462}\right)\) | \(e\left(\frac{67}{231}\right)\) | \(e\left(\frac{146}{231}\right)\) | \(e\left(\frac{67}{154}\right)\) | \(e\left(\frac{359}{462}\right)\) | \(e\left(\frac{16}{231}\right)\) | \(e\left(\frac{3}{77}\right)\) | \(e\left(\frac{134}{231}\right)\) | \(e\left(\frac{10}{231}\right)\) | \(e\left(\frac{49}{66}\right)\) |
| \(\chi_{3381}(674,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{462}\right)\) | \(e\left(\frac{17}{231}\right)\) | \(e\left(\frac{106}{231}\right)\) | \(e\left(\frac{17}{154}\right)\) | \(e\left(\frac{229}{462}\right)\) | \(e\left(\frac{173}{231}\right)\) | \(e\left(\frac{18}{77}\right)\) | \(e\left(\frac{34}{231}\right)\) | \(e\left(\frac{137}{231}\right)\) | \(e\left(\frac{41}{66}\right)\) |
| \(\chi_{3381}(695,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{383}{462}\right)\) | \(e\left(\frac{152}{231}\right)\) | \(e\left(\frac{214}{231}\right)\) | \(e\left(\frac{75}{154}\right)\) | \(e\left(\frac{349}{462}\right)\) | \(e\left(\frac{188}{231}\right)\) | \(e\left(\frac{16}{77}\right)\) | \(e\left(\frac{73}{231}\right)\) | \(e\left(\frac{2}{231}\right)\) | \(e\left(\frac{23}{66}\right)\) |
| \(\chi_{3381}(746,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{247}{462}\right)\) | \(e\left(\frac{16}{231}\right)\) | \(e\left(\frac{59}{231}\right)\) | \(e\left(\frac{93}{154}\right)\) | \(e\left(\frac{365}{462}\right)\) | \(e\left(\frac{190}{231}\right)\) | \(e\left(\frac{26}{77}\right)\) | \(e\left(\frac{32}{231}\right)\) | \(e\left(\frac{61}{231}\right)\) | \(e\left(\frac{25}{66}\right)\) |
| \(\chi_{3381}(779,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{419}{462}\right)\) | \(e\left(\frac{188}{231}\right)\) | \(e\left(\frac{58}{231}\right)\) | \(e\left(\frac{111}{154}\right)\) | \(e\left(\frac{73}{462}\right)\) | \(e\left(\frac{38}{231}\right)\) | \(e\left(\frac{36}{77}\right)\) | \(e\left(\frac{145}{231}\right)\) | \(e\left(\frac{197}{231}\right)\) | \(e\left(\frac{5}{66}\right)\) |
| \(\chi_{3381}(842,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{299}{462}\right)\) | \(e\left(\frac{68}{231}\right)\) | \(e\left(\frac{193}{231}\right)\) | \(e\left(\frac{145}{154}\right)\) | \(e\left(\frac{223}{462}\right)\) | \(e\left(\frac{230}{231}\right)\) | \(e\left(\frac{72}{77}\right)\) | \(e\left(\frac{136}{231}\right)\) | \(e\left(\frac{86}{231}\right)\) | \(e\left(\frac{65}{66}\right)\) |
| \(\chi_{3381}(872,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{337}{462}\right)\) | \(e\left(\frac{106}{231}\right)\) | \(e\left(\frac{131}{231}\right)\) | \(e\left(\frac{29}{154}\right)\) | \(e\left(\frac{137}{462}\right)\) | \(e\left(\frac{46}{231}\right)\) | \(e\left(\frac{76}{77}\right)\) | \(e\left(\frac{212}{231}\right)\) | \(e\left(\frac{202}{231}\right)\) | \(e\left(\frac{13}{66}\right)\) |