from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3332, base_ring=CyclotomicField(336))
M = H._module
chi = DirichletCharacter(H, M([168,8,21]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,3332))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3332\) | |
| Conductor: | \(3332\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(336\) | 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: | $\Q(\zeta_{336})$ |
| Fixed field: | Number field defined by a degree 336 polynomial (not computed) |
First 31 of 96 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(23\) | \(25\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3332}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{197}{336}\right)\) | \(e\left(\frac{1}{336}\right)\) | \(e\left(\frac{29}{168}\right)\) | \(e\left(\frac{299}{336}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{115}{336}\right)\) | \(e\left(\frac{1}{168}\right)\) | \(e\left(\frac{85}{112}\right)\) |
| \(\chi_{3332}(75,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{199}{336}\right)\) | \(e\left(\frac{59}{336}\right)\) | \(e\left(\frac{31}{168}\right)\) | \(e\left(\frac{169}{336}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{65}{336}\right)\) | \(e\left(\frac{59}{168}\right)\) | \(e\left(\frac{87}{112}\right)\) |
| \(\chi_{3332}(131,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{336}\right)\) | \(e\left(\frac{125}{336}\right)\) | \(e\left(\frac{97}{168}\right)\) | \(e\left(\frac{79}{336}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{263}{336}\right)\) | \(e\left(\frac{125}{168}\right)\) | \(e\left(\frac{97}{112}\right)\) |
| \(\chi_{3332}(143,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{311}{336}\right)\) | \(e\left(\frac{283}{336}\right)\) | \(e\left(\frac{143}{168}\right)\) | \(e\left(\frac{281}{336}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{289}{336}\right)\) | \(e\left(\frac{115}{168}\right)\) | \(e\left(\frac{87}{112}\right)\) |
| \(\chi_{3332}(159,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{235}{336}\right)\) | \(e\left(\frac{95}{336}\right)\) | \(e\left(\frac{67}{168}\right)\) | \(e\left(\frac{181}{336}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{173}{336}\right)\) | \(e\left(\frac{95}{168}\right)\) | \(e\left(\frac{11}{112}\right)\) |
| \(\chi_{3332}(199,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{336}\right)\) | \(e\left(\frac{253}{336}\right)\) | \(e\left(\frac{113}{168}\right)\) | \(e\left(\frac{47}{336}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{199}{336}\right)\) | \(e\left(\frac{85}{168}\right)\) | \(e\left(\frac{1}{112}\right)\) |
| \(\chi_{3332}(243,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{313}{336}\right)\) | \(e\left(\frac{5}{336}\right)\) | \(e\left(\frac{145}{168}\right)\) | \(e\left(\frac{151}{336}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{239}{336}\right)\) | \(e\left(\frac{5}{168}\right)\) | \(e\left(\frac{89}{112}\right)\) |
| \(\chi_{3332}(283,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{336}\right)\) | \(e\left(\frac{103}{336}\right)\) | \(e\left(\frac{131}{168}\right)\) | \(e\left(\frac{221}{336}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{85}{336}\right)\) | \(e\left(\frac{103}{168}\right)\) | \(e\left(\frac{19}{112}\right)\) |
| \(\chi_{3332}(299,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{127}{336}\right)\) | \(e\left(\frac{323}{336}\right)\) | \(e\left(\frac{127}{168}\right)\) | \(e\left(\frac{145}{336}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{185}{336}\right)\) | \(e\left(\frac{155}{168}\right)\) | \(e\left(\frac{15}{112}\right)\) |
| \(\chi_{3332}(311,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{137}{336}\right)\) | \(e\left(\frac{277}{336}\right)\) | \(e\left(\frac{137}{168}\right)\) | \(e\left(\frac{167}{336}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{271}{336}\right)\) | \(e\left(\frac{109}{168}\right)\) | \(e\left(\frac{25}{112}\right)\) |
| \(\chi_{3332}(367,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{191}{336}\right)\) | \(e\left(\frac{163}{336}\right)\) | \(e\left(\frac{23}{168}\right)\) | \(e\left(\frac{17}{336}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{265}{336}\right)\) | \(e\left(\frac{163}{168}\right)\) | \(e\left(\frac{79}{112}\right)\) |
| \(\chi_{3332}(439,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{336}\right)\) | \(e\left(\frac{89}{336}\right)\) | \(e\left(\frac{61}{168}\right)\) | \(e\left(\frac{67}{336}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{155}{336}\right)\) | \(e\left(\frac{89}{168}\right)\) | \(e\left(\frac{61}{112}\right)\) |
| \(\chi_{3332}(479,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{336}\right)\) | \(e\left(\frac{145}{336}\right)\) | \(e\left(\frac{5}{168}\right)\) | \(e\left(\frac{11}{336}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{211}{336}\right)\) | \(e\left(\frac{145}{168}\right)\) | \(e\left(\frac{5}{112}\right)\) |
| \(\chi_{3332}(507,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{221}{336}\right)\) | \(e\left(\frac{25}{336}\right)\) | \(e\left(\frac{53}{168}\right)\) | \(e\left(\frac{83}{336}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{187}{336}\right)\) | \(e\left(\frac{25}{168}\right)\) | \(e\left(\frac{109}{112}\right)\) |
| \(\chi_{3332}(551,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{151}{336}\right)\) | \(e\left(\frac{11}{336}\right)\) | \(e\left(\frac{151}{168}\right)\) | \(e\left(\frac{265}{336}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{257}{336}\right)\) | \(e\left(\frac{11}{168}\right)\) | \(e\left(\frac{39}{112}\right)\) |
| \(\chi_{3332}(635,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{187}{336}\right)\) | \(e\left(\frac{47}{336}\right)\) | \(e\left(\frac{19}{168}\right)\) | \(e\left(\frac{277}{336}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{29}{336}\right)\) | \(e\left(\frac{47}{168}\right)\) | \(e\left(\frac{75}{112}\right)\) |
| \(\chi_{3332}(675,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{257}{336}\right)\) | \(e\left(\frac{61}{336}\right)\) | \(e\left(\frac{89}{168}\right)\) | \(e\left(\frac{95}{336}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{295}{336}\right)\) | \(e\left(\frac{61}{168}\right)\) | \(e\left(\frac{33}{112}\right)\) |
| \(\chi_{3332}(691,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{211}{336}\right)\) | \(e\left(\frac{71}{336}\right)\) | \(e\left(\frac{43}{168}\right)\) | \(e\left(\frac{61}{336}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{101}{336}\right)\) | \(e\left(\frac{71}{168}\right)\) | \(e\left(\frac{99}{112}\right)\) |
| \(\chi_{3332}(703,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{336}\right)\) | \(e\left(\frac{319}{336}\right)\) | \(e\left(\frac{11}{168}\right)\) | \(e\left(\frac{293}{336}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{61}{336}\right)\) | \(e\left(\frac{151}{168}\right)\) | \(e\left(\frac{11}{112}\right)\) |
| \(\chi_{3332}(719,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{265}{336}\right)\) | \(e\left(\frac{293}{336}\right)\) | \(e\left(\frac{97}{168}\right)\) | \(e\left(\frac{247}{336}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{95}{336}\right)\) | \(e\left(\frac{125}{168}\right)\) | \(e\left(\frac{41}{112}\right)\) |
| \(\chi_{3332}(759,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{275}{336}\right)\) | \(e\left(\frac{247}{336}\right)\) | \(e\left(\frac{107}{168}\right)\) | \(e\left(\frac{269}{336}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{181}{336}\right)\) | \(e\left(\frac{79}{168}\right)\) | \(e\left(\frac{51}{112}\right)\) |
| \(\chi_{3332}(775,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{336}\right)\) | \(e\left(\frac{275}{336}\right)\) | \(e\left(\frac{79}{168}\right)\) | \(e\left(\frac{241}{336}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{41}{336}\right)\) | \(e\left(\frac{107}{168}\right)\) | \(e\left(\frac{79}{112}\right)\) |
| \(\chi_{3332}(787,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{281}{336}\right)\) | \(e\left(\frac{85}{336}\right)\) | \(e\left(\frac{113}{168}\right)\) | \(e\left(\frac{215}{336}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{31}{336}\right)\) | \(e\left(\frac{85}{168}\right)\) | \(e\left(\frac{57}{112}\right)\) |
| \(\chi_{3332}(843,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{335}{336}\right)\) | \(e\left(\frac{307}{336}\right)\) | \(e\left(\frac{167}{168}\right)\) | \(e\left(\frac{65}{336}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{25}{336}\right)\) | \(e\left(\frac{139}{168}\right)\) | \(e\left(\frac{111}{112}\right)\) |
| \(\chi_{3332}(887,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{85}{336}\right)\) | \(e\left(\frac{113}{336}\right)\) | \(e\left(\frac{85}{168}\right)\) | \(e\left(\frac{187}{336}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{227}{336}\right)\) | \(e\left(\frac{113}{168}\right)\) | \(e\left(\frac{85}{112}\right)\) |
| \(\chi_{3332}(915,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{336}\right)\) | \(e\left(\frac{41}{336}\right)\) | \(e\left(\frac{13}{168}\right)\) | \(e\left(\frac{163}{336}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{11}{336}\right)\) | \(e\left(\frac{41}{168}\right)\) | \(e\left(\frac{13}{112}\right)\) |
| \(\chi_{3332}(955,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{149}{336}\right)\) | \(e\left(\frac{289}{336}\right)\) | \(e\left(\frac{149}{168}\right)\) | \(e\left(\frac{59}{336}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{307}{336}\right)\) | \(e\left(\frac{121}{168}\right)\) | \(e\left(\frac{37}{112}\right)\) |
| \(\chi_{3332}(983,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{336}\right)\) | \(e\left(\frac{169}{336}\right)\) | \(e\left(\frac{29}{168}\right)\) | \(e\left(\frac{131}{336}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{283}{336}\right)\) | \(e\left(\frac{1}{168}\right)\) | \(e\left(\frac{29}{112}\right)\) |
| \(\chi_{3332}(1027,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{103}{336}\right)\) | \(e\left(\frac{299}{336}\right)\) | \(e\left(\frac{103}{168}\right)\) | \(e\left(\frac{25}{336}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{113}{336}\right)\) | \(e\left(\frac{131}{168}\right)\) | \(e\left(\frac{103}{112}\right)\) |
| \(\chi_{3332}(1083,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{336}\right)\) | \(e\left(\frac{29}{336}\right)\) | \(e\left(\frac{1}{168}\right)\) | \(e\left(\frac{271}{336}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{311}{336}\right)\) | \(e\left(\frac{29}{168}\right)\) | \(e\left(\frac{1}{112}\right)\) |
| \(\chi_{3332}(1095,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{263}{336}\right)\) | \(e\left(\frac{235}{336}\right)\) | \(e\left(\frac{95}{168}\right)\) | \(e\left(\frac{41}{336}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{145}{336}\right)\) | \(e\left(\frac{67}{168}\right)\) | \(e\left(\frac{39}{112}\right)\) |