from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(159936, base_ring=CyclotomicField(336))
M = H._module
chi = DirichletCharacter(H, M([168,315,0,200,84]))
chi.galois_orbit()
[g,chi] = znchar(Mod(115,159936))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(159936\) | |
| Conductor: | \(53312\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(336\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 53312.baz | 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 | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{159936}(115,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{151}{336}\right)\) | \(e\left(\frac{251}{336}\right)\) | \(e\left(\frac{79}{112}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{167}{168}\right)\) | \(e\left(\frac{151}{168}\right)\) | \(e\left(\frac{31}{112}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{247}{336}\right)\) | \(e\left(\frac{45}{56}\right)\) |
| \(\chi_{159936}(2299,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{336}\right)\) | \(e\left(\frac{197}{336}\right)\) | \(e\left(\frac{33}{112}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{113}{168}\right)\) | \(e\left(\frac{73}{168}\right)\) | \(e\left(\frac{81}{112}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{313}{336}\right)\) | \(e\left(\frac{11}{56}\right)\) |
| \(\chi_{159936}(3379,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{215}{336}\right)\) | \(e\left(\frac{235}{336}\right)\) | \(e\left(\frac{111}{112}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{151}{168}\right)\) | \(e\left(\frac{47}{168}\right)\) | \(e\left(\frac{79}{112}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{167}{336}\right)\) | \(e\left(\frac{37}{56}\right)\) |
| \(\chi_{159936}(5563,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{185}{336}\right)\) | \(e\left(\frac{85}{336}\right)\) | \(e\left(\frac{33}{112}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{1}{168}\right)\) | \(e\left(\frac{17}{168}\right)\) | \(e\left(\frac{81}{112}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{89}{336}\right)\) | \(e\left(\frac{11}{56}\right)\) |
| \(\chi_{159936}(5827,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{283}{336}\right)\) | \(e\left(\frac{239}{336}\right)\) | \(e\left(\frac{19}{112}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{155}{168}\right)\) | \(e\left(\frac{115}{168}\right)\) | \(e\left(\frac{67}{112}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{187}{336}\right)\) | \(e\left(\frac{25}{56}\right)\) |
| \(\chi_{159936}(8011,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{205}{336}\right)\) | \(e\left(\frac{185}{336}\right)\) | \(e\left(\frac{85}{112}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{101}{168}\right)\) | \(e\left(\frac{37}{168}\right)\) | \(e\left(\frac{5}{112}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{253}{336}\right)\) | \(e\left(\frac{47}{56}\right)\) |
| \(\chi_{159936}(9091,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{336}\right)\) | \(e\left(\frac{127}{336}\right)\) | \(e\left(\frac{19}{112}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{43}{168}\right)\) | \(e\left(\frac{59}{168}\right)\) | \(e\left(\frac{67}{112}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{299}{336}\right)\) | \(e\left(\frac{25}{56}\right)\) |
| \(\chi_{159936}(11275,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{336}\right)\) | \(e\left(\frac{313}{336}\right)\) | \(e\left(\frac{53}{112}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{61}{168}\right)\) | \(e\left(\frac{29}{168}\right)\) | \(e\left(\frac{69}{112}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{221}{336}\right)\) | \(e\left(\frac{55}{56}\right)\) |
| \(\chi_{159936}(11539,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{336}\right)\) | \(e\left(\frac{227}{336}\right)\) | \(e\left(\frac{71}{112}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{143}{168}\right)\) | \(e\left(\frac{79}{168}\right)\) | \(e\left(\frac{103}{112}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{127}{336}\right)\) | \(e\left(\frac{5}{56}\right)\) |
| \(\chi_{159936}(13723,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{336}\right)\) | \(e\left(\frac{173}{336}\right)\) | \(e\left(\frac{25}{112}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{89}{168}\right)\) | \(e\left(\frac{1}{168}\right)\) | \(e\left(\frac{41}{112}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{193}{336}\right)\) | \(e\left(\frac{27}{56}\right)\) |
| \(\chi_{159936}(14803,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{239}{336}\right)\) | \(e\left(\frac{19}{336}\right)\) | \(e\left(\frac{39}{112}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{103}{168}\right)\) | \(e\left(\frac{71}{168}\right)\) | \(e\left(\frac{55}{112}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{95}{336}\right)\) | \(e\left(\frac{13}{56}\right)\) |
| \(\chi_{159936}(16987,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{209}{336}\right)\) | \(e\left(\frac{205}{336}\right)\) | \(e\left(\frac{73}{112}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{121}{168}\right)\) | \(e\left(\frac{41}{168}\right)\) | \(e\left(\frac{57}{112}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{17}{336}\right)\) | \(e\left(\frac{43}{56}\right)\) |
| \(\chi_{159936}(17251,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{211}{336}\right)\) | \(e\left(\frac{215}{336}\right)\) | \(e\left(\frac{11}{112}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{131}{168}\right)\) | \(e\left(\frac{43}{168}\right)\) | \(e\left(\frac{27}{112}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{67}{336}\right)\) | \(e\left(\frac{41}{56}\right)\) |
| \(\chi_{159936}(20515,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{336}\right)\) | \(e\left(\frac{247}{336}\right)\) | \(e\left(\frac{59}{112}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{163}{168}\right)\) | \(e\left(\frac{83}{168}\right)\) | \(e\left(\frac{43}{112}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{227}{336}\right)\) | \(e\left(\frac{1}{56}\right)\) |
| \(\chi_{159936}(22699,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{336}\right)\) | \(e\left(\frac{97}{336}\right)\) | \(e\left(\frac{93}{112}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{13}{168}\right)\) | \(e\left(\frac{53}{168}\right)\) | \(e\left(\frac{45}{112}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{149}{336}\right)\) | \(e\left(\frac{31}{56}\right)\) |
| \(\chi_{159936}(25147,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{265}{336}\right)\) | \(e\left(\frac{149}{336}\right)\) | \(e\left(\frac{17}{112}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{65}{168}\right)\) | \(e\left(\frac{97}{168}\right)\) | \(e\left(\frac{1}{112}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{73}{336}\right)\) | \(e\left(\frac{43}{56}\right)\) |
| \(\chi_{159936}(26227,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{263}{336}\right)\) | \(e\left(\frac{139}{336}\right)\) | \(e\left(\frac{79}{112}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{55}{168}\right)\) | \(e\left(\frac{95}{168}\right)\) | \(e\left(\frac{31}{112}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{23}{336}\right)\) | \(e\left(\frac{45}{56}\right)\) |
| \(\chi_{159936}(28411,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{233}{336}\right)\) | \(e\left(\frac{325}{336}\right)\) | \(e\left(\frac{1}{112}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{73}{168}\right)\) | \(e\left(\frac{65}{168}\right)\) | \(e\left(\frac{33}{112}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{281}{336}\right)\) | \(e\left(\frac{19}{56}\right)\) |
| \(\chi_{159936}(28675,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{139}{336}\right)\) | \(e\left(\frac{191}{336}\right)\) | \(e\left(\frac{3}{112}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{107}{168}\right)\) | \(e\left(\frac{139}{168}\right)\) | \(e\left(\frac{99}{112}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{283}{336}\right)\) | \(e\left(\frac{1}{56}\right)\) |
| \(\chi_{159936}(30859,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{336}\right)\) | \(e\left(\frac{137}{336}\right)\) | \(e\left(\frac{69}{112}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{53}{168}\right)\) | \(e\left(\frac{61}{168}\right)\) | \(e\left(\frac{37}{112}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{13}{336}\right)\) | \(e\left(\frac{23}{56}\right)\) |
| \(\chi_{159936}(31939,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{336}\right)\) | \(e\left(\frac{31}{336}\right)\) | \(e\left(\frac{99}{112}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{115}{168}\right)\) | \(e\left(\frac{107}{168}\right)\) | \(e\left(\frac{19}{112}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{155}{336}\right)\) | \(e\left(\frac{33}{56}\right)\) |
| \(\chi_{159936}(34387,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{271}{336}\right)\) | \(e\left(\frac{179}{336}\right)\) | \(e\left(\frac{55}{112}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{95}{168}\right)\) | \(e\left(\frac{103}{168}\right)\) | \(e\left(\frac{23}{112}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{223}{336}\right)\) | \(e\left(\frac{37}{56}\right)\) |
| \(\chi_{159936}(36571,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{193}{336}\right)\) | \(e\left(\frac{125}{336}\right)\) | \(e\left(\frac{9}{112}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{41}{168}\right)\) | \(e\left(\frac{25}{168}\right)\) | \(e\left(\frac{73}{112}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{289}{336}\right)\) | \(e\left(\frac{3}{56}\right)\) |
| \(\chi_{159936}(39835,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{257}{336}\right)\) | \(e\left(\frac{109}{336}\right)\) | \(e\left(\frac{41}{112}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{25}{168}\right)\) | \(e\left(\frac{89}{168}\right)\) | \(e\left(\frac{9}{112}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{209}{336}\right)\) | \(e\left(\frac{51}{56}\right)\) |
| \(\chi_{159936}(40099,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{336}\right)\) | \(e\left(\frac{167}{336}\right)\) | \(e\left(\frac{107}{112}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{83}{168}\right)\) | \(e\left(\frac{67}{168}\right)\) | \(e\left(\frac{59}{112}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{163}{336}\right)\) | \(e\left(\frac{17}{56}\right)\) |
| \(\chi_{159936}(42283,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{325}{336}\right)\) | \(e\left(\frac{113}{336}\right)\) | \(e\left(\frac{61}{112}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{29}{168}\right)\) | \(e\left(\frac{157}{168}\right)\) | \(e\left(\frac{109}{112}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{229}{336}\right)\) | \(e\left(\frac{39}{56}\right)\) |
| \(\chi_{159936}(43363,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{131}{336}\right)\) | \(e\left(\frac{151}{336}\right)\) | \(e\left(\frac{27}{112}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{67}{168}\right)\) | \(e\left(\frac{131}{168}\right)\) | \(e\left(\frac{107}{112}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{83}{336}\right)\) | \(e\left(\frac{9}{56}\right)\) |
| \(\chi_{159936}(45547,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{336}\right)\) | \(e\left(\frac{1}{336}\right)\) | \(e\left(\frac{61}{112}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{85}{168}\right)\) | \(e\left(\frac{101}{168}\right)\) | \(e\left(\frac{109}{112}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{336}\right)\) | \(e\left(\frac{39}{56}\right)\) |
| \(\chi_{159936}(45811,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{199}{336}\right)\) | \(e\left(\frac{155}{336}\right)\) | \(e\left(\frac{47}{112}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{71}{168}\right)\) | \(e\left(\frac{31}{168}\right)\) | \(e\left(\frac{95}{112}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{103}{336}\right)\) | \(e\left(\frac{53}{56}\right)\) |
| \(\chi_{159936}(47995,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{121}{336}\right)\) | \(e\left(\frac{101}{336}\right)\) | \(e\left(\frac{1}{112}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{17}{168}\right)\) | \(e\left(\frac{121}{168}\right)\) | \(e\left(\frac{33}{112}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{169}{336}\right)\) | \(e\left(\frac{19}{56}\right)\) |
| \(\chi_{159936}(49075,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{311}{336}\right)\) | \(e\left(\frac{43}{336}\right)\) | \(e\left(\frac{47}{112}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{127}{168}\right)\) | \(e\left(\frac{143}{168}\right)\) | \(e\left(\frac{95}{112}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{215}{336}\right)\) | \(e\left(\frac{53}{56}\right)\) |