from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4034, base_ring=CyclotomicField(336))
M = H._module
chi = DirichletCharacter(H, M([179]))
chi.galois_orbit()
[g,chi] = znchar(Mod(27,4034))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4034\) | |
| Conductor: | \(2017\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(336\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 2017.bf | 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: | $\Q(\zeta_{336})$ |
| Fixed field: | Number field defined by a degree 336 polynomial (not computed) |
First 31 of 96 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4034}(27,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{121}{168}\right)\) | \(e\left(\frac{179}{336}\right)\) | \(e\left(\frac{109}{168}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{85}{336}\right)\) | \(e\left(\frac{59}{112}\right)\) | \(e\left(\frac{19}{336}\right)\) | \(e\left(\frac{31}{84}\right)\) |
| \(\chi_{4034}(29,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{168}\right)\) | \(e\left(\frac{313}{336}\right)\) | \(e\left(\frac{47}{168}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{143}{336}\right)\) | \(e\left(\frac{65}{112}\right)\) | \(e\left(\frac{281}{336}\right)\) | \(e\left(\frac{65}{84}\right)\) |
| \(\chi_{4034}(31,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{168}\right)\) | \(e\left(\frac{55}{336}\right)\) | \(e\left(\frac{41}{168}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{257}{336}\right)\) | \(e\left(\frac{15}{112}\right)\) | \(e\left(\frac{263}{336}\right)\) | \(e\left(\frac{71}{84}\right)\) |
| \(\chi_{4034}(33,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{95}{168}\right)\) | \(e\left(\frac{253}{336}\right)\) | \(e\left(\frac{155}{168}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{107}{336}\right)\) | \(e\left(\frac{69}{112}\right)\) | \(e\left(\frac{269}{336}\right)\) | \(e\left(\frac{41}{84}\right)\) |
| \(\chi_{4034}(63,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{168}\right)\) | \(e\left(\frac{143}{336}\right)\) | \(e\left(\frac{73}{168}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{265}{336}\right)\) | \(e\left(\frac{39}{112}\right)\) | \(e\left(\frac{79}{336}\right)\) | \(e\left(\frac{67}{84}\right)\) |
| \(\chi_{4034}(69,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{137}{168}\right)\) | \(e\left(\frac{211}{336}\right)\) | \(e\left(\frac{29}{168}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{149}{336}\right)\) | \(e\left(\frac{27}{112}\right)\) | \(e\left(\frac{227}{336}\right)\) | \(e\left(\frac{83}{84}\right)\) |
| \(\chi_{4034}(95,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{168}\right)\) | \(e\left(\frac{283}{336}\right)\) | \(e\left(\frac{101}{168}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{125}{336}\right)\) | \(e\left(\frac{67}{112}\right)\) | \(e\left(\frac{107}{336}\right)\) | \(e\left(\frac{11}{84}\right)\) |
| \(\chi_{4034}(103,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{65}{168}\right)\) | \(e\left(\frac{67}{336}\right)\) | \(e\left(\frac{53}{168}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{197}{336}\right)\) | \(e\left(\frac{59}{112}\right)\) | \(e\left(\frac{131}{336}\right)\) | \(e\left(\frac{59}{84}\right)\) |
| \(\chi_{4034}(147,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{168}\right)\) | \(e\left(\frac{107}{336}\right)\) | \(e\left(\frac{37}{168}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{109}{336}\right)\) | \(e\left(\frac{19}{112}\right)\) | \(e\left(\frac{139}{336}\right)\) | \(e\left(\frac{19}{84}\right)\) |
| \(\chi_{4034}(219,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{168}\right)\) | \(e\left(\frac{205}{336}\right)\) | \(e\left(\frac{107}{168}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{11}{336}\right)\) | \(e\left(\frac{5}{112}\right)\) | \(e\left(\frac{125}{336}\right)\) | \(e\left(\frac{5}{84}\right)\) |
| \(\chi_{4034}(235,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{168}\right)\) | \(e\left(\frac{269}{336}\right)\) | \(e\left(\frac{115}{168}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{139}{336}\right)\) | \(e\left(\frac{53}{112}\right)\) | \(e\left(\frac{205}{336}\right)\) | \(e\left(\frac{25}{84}\right)\) |
| \(\chi_{4034}(289,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{115}{168}\right)\) | \(e\left(\frac{41}{336}\right)\) | \(e\left(\frac{55}{168}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{271}{336}\right)\) | \(e\left(\frac{1}{112}\right)\) | \(e\left(\frac{25}{336}\right)\) | \(e\left(\frac{1}{84}\right)\) |
| \(\chi_{4034}(335,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{168}\right)\) | \(e\left(\frac{127}{336}\right)\) | \(e\left(\frac{113}{168}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{233}{336}\right)\) | \(e\left(\frac{55}{112}\right)\) | \(e\left(\frac{143}{336}\right)\) | \(e\left(\frac{83}{84}\right)\) |
| \(\chi_{4034}(343,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{168}\right)\) | \(e\left(\frac{71}{336}\right)\) | \(e\left(\frac{1}{168}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{289}{336}\right)\) | \(e\left(\frac{111}{112}\right)\) | \(e\left(\frac{199}{336}\right)\) | \(e\left(\frac{55}{84}\right)\) |
| \(\chi_{4034}(383,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{151}{168}\right)\) | \(e\left(\frac{197}{336}\right)\) | \(e\left(\frac{43}{168}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{163}{336}\right)\) | \(e\left(\frac{13}{112}\right)\) | \(e\left(\frac{325}{336}\right)\) | \(e\left(\frac{13}{84}\right)\) |
| \(\chi_{4034}(489,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{168}\right)\) | \(e\left(\frac{83}{336}\right)\) | \(e\left(\frac{13}{168}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{229}{336}\right)\) | \(e\left(\frac{43}{112}\right)\) | \(e\left(\frac{67}{336}\right)\) | \(e\left(\frac{43}{84}\right)\) |
| \(\chi_{4034}(511,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{168}\right)\) | \(e\left(\frac{169}{336}\right)\) | \(e\left(\frac{71}{168}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{191}{336}\right)\) | \(e\left(\frac{97}{112}\right)\) | \(e\left(\frac{185}{336}\right)\) | \(e\left(\frac{41}{84}\right)\) |
| \(\chi_{4034}(541,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{168}\right)\) | \(e\left(\frac{97}{336}\right)\) | \(e\left(\frac{167}{168}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{215}{336}\right)\) | \(e\left(\frac{57}{112}\right)\) | \(e\left(\frac{305}{336}\right)\) | \(e\left(\frac{29}{84}\right)\) |
| \(\chi_{4034}(617,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{149}{168}\right)\) | \(e\left(\frac{319}{336}\right)\) | \(e\left(\frac{137}{168}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{281}{336}\right)\) | \(e\left(\frac{87}{112}\right)\) | \(e\left(\frac{47}{336}\right)\) | \(e\left(\frac{59}{84}\right)\) |
| \(\chi_{4034}(677,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{163}{168}\right)\) | \(e\left(\frac{305}{336}\right)\) | \(e\left(\frac{151}{168}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{295}{336}\right)\) | \(e\left(\frac{73}{112}\right)\) | \(e\left(\frac{145}{336}\right)\) | \(e\left(\frac{73}{84}\right)\) |
| \(\chi_{4034}(747,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{168}\right)\) | \(e\left(\frac{325}{336}\right)\) | \(e\left(\frac{59}{168}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{83}{336}\right)\) | \(e\left(\frac{109}{112}\right)\) | \(e\left(\frac{149}{336}\right)\) | \(e\left(\frac{53}{84}\right)\) |
| \(\chi_{4034}(921,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{168}\right)\) | \(e\left(\frac{299}{336}\right)\) | \(e\left(\frac{61}{168}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{157}{336}\right)\) | \(e\left(\frac{51}{112}\right)\) | \(e\left(\frac{43}{336}\right)\) | \(e\left(\frac{79}{84}\right)\) |
| \(\chi_{4034}(969,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{168}\right)\) | \(e\left(\frac{307}{336}\right)\) | \(e\left(\frac{125}{168}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{5}{336}\right)\) | \(e\left(\frac{43}{112}\right)\) | \(e\left(\frac{179}{336}\right)\) | \(e\left(\frac{71}{84}\right)\) |
| \(\chi_{4034}(971,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{157}{168}\right)\) | \(e\left(\frac{335}{336}\right)\) | \(e\left(\frac{97}{168}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{313}{336}\right)\) | \(e\left(\frac{71}{112}\right)\) | \(e\left(\frac{319}{336}\right)\) | \(e\left(\frac{43}{84}\right)\) |
| \(\chi_{4034}(1009,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{168}\right)\) | \(e\left(\frac{331}{336}\right)\) | \(e\left(\frac{149}{168}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{221}{336}\right)\) | \(e\left(\frac{19}{112}\right)\) | \(e\left(\frac{251}{336}\right)\) | \(e\left(\frac{47}{84}\right)\) |
| \(\chi_{4034}(1041,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{168}\right)\) | \(e\left(\frac{113}{336}\right)\) | \(e\left(\frac{127}{168}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{247}{336}\right)\) | \(e\left(\frac{41}{112}\right)\) | \(e\left(\frac{241}{336}\right)\) | \(e\left(\frac{13}{84}\right)\) |
| \(\chi_{4034}(1123,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{168}\right)\) | \(e\left(\frac{79}{336}\right)\) | \(e\left(\frac{65}{168}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{137}{336}\right)\) | \(e\left(\frac{103}{112}\right)\) | \(e\left(\frac{335}{336}\right)\) | \(e\left(\frac{47}{84}\right)\) |
| \(\chi_{4034}(1141,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{168}\right)\) | \(e\left(\frac{47}{336}\right)\) | \(e\left(\frac{145}{168}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{73}{336}\right)\) | \(e\left(\frac{23}{112}\right)\) | \(e\left(\frac{127}{336}\right)\) | \(e\left(\frac{79}{84}\right)\) |
| \(\chi_{4034}(1165,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{143}{168}\right)\) | \(e\left(\frac{13}{336}\right)\) | \(e\left(\frac{83}{168}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{299}{336}\right)\) | \(e\left(\frac{85}{112}\right)\) | \(e\left(\frac{221}{336}\right)\) | \(e\left(\frac{29}{84}\right)\) |
| \(\chi_{4034}(1277,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{168}\right)\) | \(e\left(\frac{109}{336}\right)\) | \(e\left(\frac{11}{168}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{155}{336}\right)\) | \(e\left(\frac{101}{112}\right)\) | \(e\left(\frac{173}{336}\right)\) | \(e\left(\frac{17}{84}\right)\) |
| \(\chi_{4034}(1373,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{168}\right)\) | \(e\left(\frac{17}{336}\right)\) | \(e\left(\frac{31}{168}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{55}{336}\right)\) | \(e\left(\frac{25}{112}\right)\) | \(e\left(\frac{289}{336}\right)\) | \(e\left(\frac{25}{84}\right)\) |