from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(16335, base_ring=CyclotomicField(198))
M = H._module
chi = DirichletCharacter(H, M([11,0,18]))
chi.galois_orbit()
[g,chi] = znchar(Mod(56,16335))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(16335\) | |
| Conductor: | \(3267\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(198\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 3267.bn | 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: | $\Q(\zeta_{99})$ |
| Fixed field: | Number field defined by a degree 198 polynomial (not computed) |
First 31 of 60 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{16335}(56,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{198}\right)\) | \(e\left(\frac{29}{99}\right)\) | \(e\left(\frac{52}{99}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{62}{99}\right)\) | \(e\left(\frac{133}{198}\right)\) | \(e\left(\frac{58}{99}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{193}{198}\right)\) |
| \(\chi_{16335}(221,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{127}{198}\right)\) | \(e\left(\frac{28}{99}\right)\) | \(e\left(\frac{98}{99}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{94}{99}\right)\) | \(e\left(\frac{125}{198}\right)\) | \(e\left(\frac{56}{99}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{101}{198}\right)\) |
| \(\chi_{16335}(551,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{125}{198}\right)\) | \(e\left(\frac{26}{99}\right)\) | \(e\left(\frac{91}{99}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{59}{99}\right)\) | \(e\left(\frac{109}{198}\right)\) | \(e\left(\frac{52}{99}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{115}{198}\right)\) |
| \(\chi_{16335}(716,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{198}\right)\) | \(e\left(\frac{25}{99}\right)\) | \(e\left(\frac{38}{99}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{91}{99}\right)\) | \(e\left(\frac{101}{198}\right)\) | \(e\left(\frac{50}{99}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{23}{198}\right)\) |
| \(\chi_{16335}(1046,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{198}\right)\) | \(e\left(\frac{23}{99}\right)\) | \(e\left(\frac{31}{99}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{56}{99}\right)\) | \(e\left(\frac{85}{198}\right)\) | \(e\left(\frac{46}{99}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{37}{198}\right)\) |
| \(\chi_{16335}(1541,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{119}{198}\right)\) | \(e\left(\frac{20}{99}\right)\) | \(e\left(\frac{70}{99}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{53}{99}\right)\) | \(e\left(\frac{61}{198}\right)\) | \(e\left(\frac{40}{99}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{157}{198}\right)\) |
| \(\chi_{16335}(1706,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{198}\right)\) | \(e\left(\frac{19}{99}\right)\) | \(e\left(\frac{17}{99}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{85}{99}\right)\) | \(e\left(\frac{53}{198}\right)\) | \(e\left(\frac{38}{99}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{65}{198}\right)\) |
| \(\chi_{16335}(2036,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{198}\right)\) | \(e\left(\frac{17}{99}\right)\) | \(e\left(\frac{10}{99}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{50}{99}\right)\) | \(e\left(\frac{37}{198}\right)\) | \(e\left(\frac{34}{99}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{79}{198}\right)\) |
| \(\chi_{16335}(2201,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{115}{198}\right)\) | \(e\left(\frac{16}{99}\right)\) | \(e\left(\frac{56}{99}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{82}{99}\right)\) | \(e\left(\frac{29}{198}\right)\) | \(e\left(\frac{32}{99}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{185}{198}\right)\) |
| \(\chi_{16335}(2531,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{198}\right)\) | \(e\left(\frac{14}{99}\right)\) | \(e\left(\frac{49}{99}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{47}{99}\right)\) | \(e\left(\frac{13}{198}\right)\) | \(e\left(\frac{28}{99}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{1}{198}\right)\) |
| \(\chi_{16335}(2696,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{198}\right)\) | \(e\left(\frac{13}{99}\right)\) | \(e\left(\frac{95}{99}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{79}{99}\right)\) | \(e\left(\frac{5}{198}\right)\) | \(e\left(\frac{26}{99}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{107}{198}\right)\) |
| \(\chi_{16335}(3191,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{198}\right)\) | \(e\left(\frac{10}{99}\right)\) | \(e\left(\frac{35}{99}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{76}{99}\right)\) | \(e\left(\frac{179}{198}\right)\) | \(e\left(\frac{20}{99}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{29}{198}\right)\) |
| \(\chi_{16335}(3521,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{198}\right)\) | \(e\left(\frac{8}{99}\right)\) | \(e\left(\frac{28}{99}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{41}{99}\right)\) | \(e\left(\frac{163}{198}\right)\) | \(e\left(\frac{16}{99}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{43}{198}\right)\) |
| \(\chi_{16335}(3686,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{198}\right)\) | \(e\left(\frac{7}{99}\right)\) | \(e\left(\frac{74}{99}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{73}{99}\right)\) | \(e\left(\frac{155}{198}\right)\) | \(e\left(\frac{14}{99}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{149}{198}\right)\) |
| \(\chi_{16335}(4016,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{198}\right)\) | \(e\left(\frac{5}{99}\right)\) | \(e\left(\frac{67}{99}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{38}{99}\right)\) | \(e\left(\frac{139}{198}\right)\) | \(e\left(\frac{10}{99}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{163}{198}\right)\) |
| \(\chi_{16335}(4181,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{103}{198}\right)\) | \(e\left(\frac{4}{99}\right)\) | \(e\left(\frac{14}{99}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{70}{99}\right)\) | \(e\left(\frac{131}{198}\right)\) | \(e\left(\frac{8}{99}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{71}{198}\right)\) |
| \(\chi_{16335}(4511,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{198}\right)\) | \(e\left(\frac{2}{99}\right)\) | \(e\left(\frac{7}{99}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{35}{99}\right)\) | \(e\left(\frac{115}{198}\right)\) | \(e\left(\frac{4}{99}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{85}{198}\right)\) |
| \(\chi_{16335}(4676,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{198}\right)\) | \(e\left(\frac{1}{99}\right)\) | \(e\left(\frac{53}{99}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{67}{99}\right)\) | \(e\left(\frac{107}{198}\right)\) | \(e\left(\frac{2}{99}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{191}{198}\right)\) |
| \(\chi_{16335}(5006,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{197}{198}\right)\) | \(e\left(\frac{98}{99}\right)\) | \(e\left(\frac{46}{99}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{32}{99}\right)\) | \(e\left(\frac{91}{198}\right)\) | \(e\left(\frac{97}{99}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{7}{198}\right)\) |
| \(\chi_{16335}(5171,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{198}\right)\) | \(e\left(\frac{97}{99}\right)\) | \(e\left(\frac{92}{99}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{64}{99}\right)\) | \(e\left(\frac{83}{198}\right)\) | \(e\left(\frac{95}{99}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{113}{198}\right)\) |
| \(\chi_{16335}(5501,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{95}{198}\right)\) | \(e\left(\frac{95}{99}\right)\) | \(e\left(\frac{85}{99}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{29}{99}\right)\) | \(e\left(\frac{67}{198}\right)\) | \(e\left(\frac{91}{99}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{127}{198}\right)\) |
| \(\chi_{16335}(5666,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{193}{198}\right)\) | \(e\left(\frac{94}{99}\right)\) | \(e\left(\frac{32}{99}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{61}{99}\right)\) | \(e\left(\frac{59}{198}\right)\) | \(e\left(\frac{89}{99}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{35}{198}\right)\) |
| \(\chi_{16335}(5996,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{191}{198}\right)\) | \(e\left(\frac{92}{99}\right)\) | \(e\left(\frac{25}{99}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{26}{99}\right)\) | \(e\left(\frac{43}{198}\right)\) | \(e\left(\frac{85}{99}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{49}{198}\right)\) |
| \(\chi_{16335}(6161,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{198}\right)\) | \(e\left(\frac{91}{99}\right)\) | \(e\left(\frac{71}{99}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{58}{99}\right)\) | \(e\left(\frac{35}{198}\right)\) | \(e\left(\frac{83}{99}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{155}{198}\right)\) |
| \(\chi_{16335}(6491,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{198}\right)\) | \(e\left(\frac{89}{99}\right)\) | \(e\left(\frac{64}{99}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{23}{99}\right)\) | \(e\left(\frac{19}{198}\right)\) | \(e\left(\frac{79}{99}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{169}{198}\right)\) |
| \(\chi_{16335}(6986,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{185}{198}\right)\) | \(e\left(\frac{86}{99}\right)\) | \(e\left(\frac{4}{99}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{20}{99}\right)\) | \(e\left(\frac{193}{198}\right)\) | \(e\left(\frac{73}{99}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{91}{198}\right)\) |
| \(\chi_{16335}(7151,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{85}{198}\right)\) | \(e\left(\frac{85}{99}\right)\) | \(e\left(\frac{50}{99}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{52}{99}\right)\) | \(e\left(\frac{185}{198}\right)\) | \(e\left(\frac{71}{99}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{197}{198}\right)\) |
| \(\chi_{16335}(7481,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{198}\right)\) | \(e\left(\frac{83}{99}\right)\) | \(e\left(\frac{43}{99}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{17}{99}\right)\) | \(e\left(\frac{169}{198}\right)\) | \(e\left(\frac{67}{99}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{13}{198}\right)\) |
| \(\chi_{16335}(7646,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{181}{198}\right)\) | \(e\left(\frac{82}{99}\right)\) | \(e\left(\frac{89}{99}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{49}{99}\right)\) | \(e\left(\frac{161}{198}\right)\) | \(e\left(\frac{65}{99}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{119}{198}\right)\) |
| \(\chi_{16335}(7976,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{179}{198}\right)\) | \(e\left(\frac{80}{99}\right)\) | \(e\left(\frac{82}{99}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{14}{99}\right)\) | \(e\left(\frac{145}{198}\right)\) | \(e\left(\frac{61}{99}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{133}{198}\right)\) |
| \(\chi_{16335}(8141,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{198}\right)\) | \(e\left(\frac{79}{99}\right)\) | \(e\left(\frac{29}{99}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{46}{99}\right)\) | \(e\left(\frac{137}{198}\right)\) | \(e\left(\frac{59}{99}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{41}{198}\right)\) |