from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1297, base_ring=CyclotomicField(1296))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(10,1297))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1297\) | |
| Conductor: | \(1297\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1296\) | 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: | $\Q(\zeta_{1296})$ |
| Fixed field: | Number field defined by a degree 1296 polynomial (not computed) |
First 31 of 432 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1297}(10,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{239}{648}\right)\) | \(e\left(\frac{1}{162}\right)\) | \(e\left(\frac{239}{324}\right)\) | \(e\left(\frac{91}{144}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{115}{648}\right)\) | \(e\left(\frac{23}{216}\right)\) | \(e\left(\frac{1}{81}\right)\) | \(e\left(\frac{1}{1296}\right)\) | \(e\left(\frac{209}{432}\right)\) |
| \(\chi_{1297}(15,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{648}\right)\) | \(e\left(\frac{17}{162}\right)\) | \(e\left(\frac{13}{324}\right)\) | \(e\left(\frac{89}{144}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{497}{648}\right)\) | \(e\left(\frac{13}{216}\right)\) | \(e\left(\frac{17}{81}\right)\) | \(e\left(\frac{827}{1296}\right)\) | \(e\left(\frac{43}{432}\right)\) |
| \(\chi_{1297}(17,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{301}{648}\right)\) | \(e\left(\frac{107}{162}\right)\) | \(e\left(\frac{301}{324}\right)\) | \(e\left(\frac{89}{144}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{641}{648}\right)\) | \(e\left(\frac{85}{216}\right)\) | \(e\left(\frac{26}{81}\right)\) | \(e\left(\frac{107}{1296}\right)\) | \(e\left(\frac{331}{432}\right)\) |
| \(\chi_{1297}(20,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{433}{648}\right)\) | \(e\left(\frac{155}{162}\right)\) | \(e\left(\frac{109}{324}\right)\) | \(e\left(\frac{101}{144}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{648}\right)\) | \(e\left(\frac{1}{216}\right)\) | \(e\left(\frac{74}{81}\right)\) | \(e\left(\frac{479}{1296}\right)\) | \(e\left(\frac{319}{432}\right)\) |
| \(\chi_{1297}(22,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{359}{648}\right)\) | \(e\left(\frac{133}{162}\right)\) | \(e\left(\frac{35}{324}\right)\) | \(e\left(\frac{43}{144}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{67}{648}\right)\) | \(e\left(\frac{143}{216}\right)\) | \(e\left(\frac{52}{81}\right)\) | \(e\left(\frac{1105}{1296}\right)\) | \(e\left(\frac{257}{432}\right)\) |
| \(\chi_{1297}(29,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{601}{648}\right)\) | \(e\left(\frac{113}{162}\right)\) | \(e\left(\frac{277}{324}\right)\) | \(e\left(\frac{77}{144}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{197}{648}\right)\) | \(e\left(\frac{169}{216}\right)\) | \(e\left(\frac{32}{81}\right)\) | \(e\left(\frac{599}{1296}\right)\) | \(e\left(\frac{343}{432}\right)\) |
| \(\chi_{1297}(31,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{641}{648}\right)\) | \(e\left(\frac{103}{162}\right)\) | \(e\left(\frac{317}{324}\right)\) | \(e\left(\frac{13}{144}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{181}{648}\right)\) | \(e\left(\frac{209}{216}\right)\) | \(e\left(\frac{22}{81}\right)\) | \(e\left(\frac{103}{1296}\right)\) | \(e\left(\frac{359}{432}\right)\) |
| \(\chi_{1297}(33,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{133}{648}\right)\) | \(e\left(\frac{149}{162}\right)\) | \(e\left(\frac{133}{324}\right)\) | \(e\left(\frac{41}{144}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{449}{648}\right)\) | \(e\left(\frac{133}{216}\right)\) | \(e\left(\frac{68}{81}\right)\) | \(e\left(\frac{635}{1296}\right)\) | \(e\left(\frac{91}{432}\right)\) |
| \(\chi_{1297}(35,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{583}{648}\right)\) | \(e\left(\frac{77}{162}\right)\) | \(e\left(\frac{259}{324}\right)\) | \(e\left(\frac{131}{144}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{107}{648}\right)\) | \(e\left(\frac{151}{216}\right)\) | \(e\left(\frac{77}{81}\right)\) | \(e\left(\frac{1049}{1296}\right)\) | \(e\left(\frac{217}{432}\right)\) |
| \(\chi_{1297}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{259}{648}\right)\) | \(e\left(\frac{77}{162}\right)\) | \(e\left(\frac{259}{324}\right)\) | \(e\left(\frac{95}{144}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{431}{648}\right)\) | \(e\left(\frac{43}{216}\right)\) | \(e\left(\frac{77}{81}\right)\) | \(e\left(\frac{77}{1296}\right)\) | \(e\left(\frac{109}{432}\right)\) |
| \(\chi_{1297}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{331}{648}\right)\) | \(e\left(\frac{59}{162}\right)\) | \(e\left(\frac{7}{324}\right)\) | \(e\left(\frac{23}{144}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{143}{648}\right)\) | \(e\left(\frac{115}{216}\right)\) | \(e\left(\frac{59}{81}\right)\) | \(e\left(\frac{869}{1296}\right)\) | \(e\left(\frac{181}{432}\right)\) |
| \(\chi_{1297}(44,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{553}{648}\right)\) | \(e\left(\frac{125}{162}\right)\) | \(e\left(\frac{229}{324}\right)\) | \(e\left(\frac{53}{144}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{605}{648}\right)\) | \(e\left(\frac{121}{216}\right)\) | \(e\left(\frac{44}{81}\right)\) | \(e\left(\frac{287}{1296}\right)\) | \(e\left(\frac{367}{432}\right)\) |
| \(\chi_{1297}(45,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{629}{648}\right)\) | \(e\left(\frac{25}{162}\right)\) | \(e\left(\frac{305}{324}\right)\) | \(e\left(\frac{97}{144}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{121}{648}\right)\) | \(e\left(\frac{197}{216}\right)\) | \(e\left(\frac{25}{81}\right)\) | \(e\left(\frac{835}{1296}\right)\) | \(e\left(\frac{419}{432}\right)\) |
| \(\chi_{1297}(51,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{269}{648}\right)\) | \(e\left(\frac{115}{162}\right)\) | \(e\left(\frac{269}{324}\right)\) | \(e\left(\frac{97}{144}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{265}{648}\right)\) | \(e\left(\frac{53}{216}\right)\) | \(e\left(\frac{34}{81}\right)\) | \(e\left(\frac{115}{1296}\right)\) | \(e\left(\frac{275}{432}\right)\) |
| \(\chi_{1297}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{289}{648}\right)\) | \(e\left(\frac{29}{162}\right)\) | \(e\left(\frac{289}{324}\right)\) | \(e\left(\frac{101}{144}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{581}{648}\right)\) | \(e\left(\frac{73}{216}\right)\) | \(e\left(\frac{29}{81}\right)\) | \(e\left(\frac{191}{1296}\right)\) | \(e\left(\frac{175}{432}\right)\) |
| \(\chi_{1297}(60,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{401}{648}\right)\) | \(e\left(\frac{1}{162}\right)\) | \(e\left(\frac{77}{324}\right)\) | \(e\left(\frac{109}{144}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{277}{648}\right)\) | \(e\left(\frac{185}{216}\right)\) | \(e\left(\frac{1}{81}\right)\) | \(e\left(\frac{487}{1296}\right)\) | \(e\left(\frac{263}{432}\right)\) |
| \(\chi_{1297}(62,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{187}{648}\right)\) | \(e\left(\frac{95}{162}\right)\) | \(e\left(\frac{187}{324}\right)\) | \(e\left(\frac{23}{144}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{71}{648}\right)\) | \(e\left(\frac{187}{216}\right)\) | \(e\left(\frac{14}{81}\right)\) | \(e\left(\frac{581}{1296}\right)\) | \(e\left(\frac{37}{432}\right)\) |
| \(\chi_{1297}(68,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{648}\right)\) | \(e\left(\frac{91}{162}\right)\) | \(e\left(\frac{41}{324}\right)\) | \(e\left(\frac{109}{144}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{421}{648}\right)\) | \(e\left(\frac{41}{216}\right)\) | \(e\left(\frac{10}{81}\right)\) | \(e\left(\frac{1063}{1296}\right)\) | \(e\left(\frac{119}{432}\right)\) |
| \(\chi_{1297}(77,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{648}\right)\) | \(e\left(\frac{47}{162}\right)\) | \(e\left(\frac{55}{324}\right)\) | \(e\left(\frac{83}{144}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{59}{648}\right)\) | \(e\left(\frac{55}{216}\right)\) | \(e\left(\frac{47}{81}\right)\) | \(e\left(\frac{857}{1296}\right)\) | \(e\left(\frac{265}{432}\right)\) |
| \(\chi_{1297}(79,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{648}\right)\) | \(e\left(\frac{19}{162}\right)\) | \(e\left(\frac{5}{324}\right)\) | \(e\left(\frac{73}{144}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{241}{648}\right)\) | \(e\left(\frac{5}{216}\right)\) | \(e\left(\frac{19}{81}\right)\) | \(e\left(\frac{667}{1296}\right)\) | \(e\left(\frac{299}{432}\right)\) |
| \(\chi_{1297}(80,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{173}{648}\right)\) | \(e\left(\frac{139}{162}\right)\) | \(e\left(\frac{173}{324}\right)\) | \(e\left(\frac{121}{144}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{433}{648}\right)\) | \(e\left(\frac{173}{216}\right)\) | \(e\left(\frac{58}{81}\right)\) | \(e\left(\frac{139}{1296}\right)\) | \(e\left(\frac{107}{432}\right)\) |
| \(\chi_{1297}(82,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{533}{648}\right)\) | \(e\left(\frac{49}{162}\right)\) | \(e\left(\frac{209}{324}\right)\) | \(e\left(\frac{49}{144}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{289}{648}\right)\) | \(e\left(\frac{101}{216}\right)\) | \(e\left(\frac{49}{81}\right)\) | \(e\left(\frac{211}{1296}\right)\) | \(e\left(\frac{35}{432}\right)\) |
| \(\chi_{1297}(87,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{569}{648}\right)\) | \(e\left(\frac{121}{162}\right)\) | \(e\left(\frac{245}{324}\right)\) | \(e\left(\frac{85}{144}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{469}{648}\right)\) | \(e\left(\frac{137}{216}\right)\) | \(e\left(\frac{40}{81}\right)\) | \(e\left(\frac{607}{1296}\right)\) | \(e\left(\frac{287}{432}\right)\) |
| \(\chi_{1297}(89,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{185}{648}\right)\) | \(e\left(\frac{55}{162}\right)\) | \(e\left(\frac{185}{324}\right)\) | \(e\left(\frac{109}{144}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{493}{648}\right)\) | \(e\left(\frac{185}{216}\right)\) | \(e\left(\frac{55}{81}\right)\) | \(e\left(\frac{55}{1296}\right)\) | \(e\left(\frac{263}{432}\right)\) |
| \(\chi_{1297}(90,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{175}{648}\right)\) | \(e\left(\frac{17}{162}\right)\) | \(e\left(\frac{175}{324}\right)\) | \(e\left(\frac{107}{144}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{11}{648}\right)\) | \(e\left(\frac{175}{216}\right)\) | \(e\left(\frac{17}{81}\right)\) | \(e\left(\frac{17}{1296}\right)\) | \(e\left(\frac{97}{432}\right)\) |
| \(\chi_{1297}(95,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{648}\right)\) | \(e\left(\frac{89}{162}\right)\) | \(e\left(\frac{49}{324}\right)\) | \(e\left(\frac{125}{144}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{29}{648}\right)\) | \(e\left(\frac{49}{216}\right)\) | \(e\left(\frac{8}{81}\right)\) | \(e\left(\frac{1223}{1296}\right)\) | \(e\left(\frac{295}{432}\right)\) |
| \(\chi_{1297}(99,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{648}\right)\) | \(e\left(\frac{157}{162}\right)\) | \(e\left(\frac{101}{324}\right)\) | \(e\left(\frac{49}{144}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{73}{648}\right)\) | \(e\left(\frac{101}{216}\right)\) | \(e\left(\frac{76}{81}\right)\) | \(e\left(\frac{643}{1296}\right)\) | \(e\left(\frac{35}{432}\right)\) |
| \(\chi_{1297}(102,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{463}{648}\right)\) | \(e\left(\frac{107}{162}\right)\) | \(e\left(\frac{139}{324}\right)\) | \(e\left(\frac{107}{144}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{155}{648}\right)\) | \(e\left(\frac{31}{216}\right)\) | \(e\left(\frac{26}{81}\right)\) | \(e\left(\frac{593}{1296}\right)\) | \(e\left(\frac{385}{432}\right)\) |
| \(\chi_{1297}(105,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{551}{648}\right)\) | \(e\left(\frac{85}{162}\right)\) | \(e\left(\frac{227}{324}\right)\) | \(e\left(\frac{139}{144}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{379}{648}\right)\) | \(e\left(\frac{119}{216}\right)\) | \(e\left(\frac{4}{81}\right)\) | \(e\left(\frac{1057}{1296}\right)\) | \(e\left(\frac{161}{432}\right)\) |
| \(\chi_{1297}(109,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{457}{648}\right)\) | \(e\left(\frac{149}{162}\right)\) | \(e\left(\frac{133}{324}\right)\) | \(e\left(\frac{5}{144}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{125}{648}\right)\) | \(e\left(\frac{25}{216}\right)\) | \(e\left(\frac{68}{81}\right)\) | \(e\left(\frac{959}{1296}\right)\) | \(e\left(\frac{415}{432}\right)\) |
| \(\chi_{1297}(111,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{227}{648}\right)\) | \(e\left(\frac{85}{162}\right)\) | \(e\left(\frac{227}{324}\right)\) | \(e\left(\frac{103}{144}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{55}{648}\right)\) | \(e\left(\frac{11}{216}\right)\) | \(e\left(\frac{4}{81}\right)\) | \(e\left(\frac{85}{1296}\right)\) | \(e\left(\frac{53}{432}\right)\) |