from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3267, base_ring=CyclotomicField(198))
M = H._module
chi = DirichletCharacter(H, M([44,45]))
chi.galois_orbit()
[g,chi] = znchar(Mod(43,3267))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3267\) | |
| Conductor: | \(3267\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(198\) | 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: | $\Q(\zeta_{99})$ |
| Fixed field: | Number field defined by a degree 198 polynomial (not computed) |
First 31 of 60 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3267}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{198}\right)\) | \(e\left(\frac{89}{99}\right)\) | \(e\left(\frac{92}{99}\right)\) | \(e\left(\frac{29}{198}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{145}{198}\right)\) | \(e\left(\frac{59}{99}\right)\) | \(e\left(\frac{79}{99}\right)\) | \(e\left(\frac{31}{66}\right)\) |
| \(\chi_{3267}(76,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{198}\right)\) | \(e\left(\frac{10}{99}\right)\) | \(e\left(\frac{7}{99}\right)\) | \(e\left(\frac{169}{198}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{53}{198}\right)\) | \(e\left(\frac{40}{99}\right)\) | \(e\left(\frac{20}{99}\right)\) | \(e\left(\frac{35}{66}\right)\) |
| \(\chi_{3267}(142,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{149}{198}\right)\) | \(e\left(\frac{50}{99}\right)\) | \(e\left(\frac{35}{99}\right)\) | \(e\left(\frac{53}{198}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{67}{198}\right)\) | \(e\left(\frac{2}{99}\right)\) | \(e\left(\frac{1}{99}\right)\) | \(e\left(\frac{43}{66}\right)\) |
| \(\chi_{3267}(175,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{169}{198}\right)\) | \(e\left(\frac{70}{99}\right)\) | \(e\left(\frac{49}{99}\right)\) | \(e\left(\frac{193}{198}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{173}{198}\right)\) | \(e\left(\frac{82}{99}\right)\) | \(e\left(\frac{41}{99}\right)\) | \(e\left(\frac{47}{66}\right)\) |
| \(\chi_{3267}(274,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{198}\right)\) | \(e\left(\frac{31}{99}\right)\) | \(e\left(\frac{91}{99}\right)\) | \(e\left(\frac{19}{198}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{95}{198}\right)\) | \(e\left(\frac{25}{99}\right)\) | \(e\left(\frac{62}{99}\right)\) | \(e\left(\frac{59}{66}\right)\) |
| \(\chi_{3267}(340,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{198}\right)\) | \(e\left(\frac{71}{99}\right)\) | \(e\left(\frac{20}{99}\right)\) | \(e\left(\frac{101}{198}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{109}{198}\right)\) | \(e\left(\frac{86}{99}\right)\) | \(e\left(\frac{43}{99}\right)\) | \(e\left(\frac{1}{66}\right)\) |
| \(\chi_{3267}(373,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{198}\right)\) | \(e\left(\frac{91}{99}\right)\) | \(e\left(\frac{34}{99}\right)\) | \(e\left(\frac{43}{198}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{17}{198}\right)\) | \(e\left(\frac{67}{99}\right)\) | \(e\left(\frac{83}{99}\right)\) | \(e\left(\frac{5}{66}\right)\) |
| \(\chi_{3267}(439,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{198}\right)\) | \(e\left(\frac{32}{99}\right)\) | \(e\left(\frac{62}{99}\right)\) | \(e\left(\frac{125}{198}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{31}{198}\right)\) | \(e\left(\frac{29}{99}\right)\) | \(e\left(\frac{64}{99}\right)\) | \(e\left(\frac{13}{66}\right)\) |
| \(\chi_{3267}(472,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{151}{198}\right)\) | \(e\left(\frac{52}{99}\right)\) | \(e\left(\frac{76}{99}\right)\) | \(e\left(\frac{67}{198}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{137}{198}\right)\) | \(e\left(\frac{10}{99}\right)\) | \(e\left(\frac{5}{99}\right)\) | \(e\left(\frac{17}{66}\right)\) |
| \(\chi_{3267}(538,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{191}{198}\right)\) | \(e\left(\frac{92}{99}\right)\) | \(e\left(\frac{5}{99}\right)\) | \(e\left(\frac{149}{198}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{151}{198}\right)\) | \(e\left(\frac{71}{99}\right)\) | \(e\left(\frac{85}{99}\right)\) | \(e\left(\frac{25}{66}\right)\) |
| \(\chi_{3267}(571,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{198}\right)\) | \(e\left(\frac{13}{99}\right)\) | \(e\left(\frac{19}{99}\right)\) | \(e\left(\frac{91}{198}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{59}{198}\right)\) | \(e\left(\frac{52}{99}\right)\) | \(e\left(\frac{26}{99}\right)\) | \(e\left(\frac{29}{66}\right)\) |
| \(\chi_{3267}(637,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{198}\right)\) | \(e\left(\frac{53}{99}\right)\) | \(e\left(\frac{47}{99}\right)\) | \(e\left(\frac{173}{198}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{73}{198}\right)\) | \(e\left(\frac{14}{99}\right)\) | \(e\left(\frac{7}{99}\right)\) | \(e\left(\frac{37}{66}\right)\) |
| \(\chi_{3267}(670,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{198}\right)\) | \(e\left(\frac{73}{99}\right)\) | \(e\left(\frac{61}{99}\right)\) | \(e\left(\frac{115}{198}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{179}{198}\right)\) | \(e\left(\frac{94}{99}\right)\) | \(e\left(\frac{47}{99}\right)\) | \(e\left(\frac{41}{66}\right)\) |
| \(\chi_{3267}(736,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{198}\right)\) | \(e\left(\frac{14}{99}\right)\) | \(e\left(\frac{89}{99}\right)\) | \(e\left(\frac{197}{198}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{193}{198}\right)\) | \(e\left(\frac{56}{99}\right)\) | \(e\left(\frac{28}{99}\right)\) | \(e\left(\frac{49}{66}\right)\) |
| \(\chi_{3267}(769,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{133}{198}\right)\) | \(e\left(\frac{34}{99}\right)\) | \(e\left(\frac{4}{99}\right)\) | \(e\left(\frac{139}{198}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{101}{198}\right)\) | \(e\left(\frac{37}{99}\right)\) | \(e\left(\frac{68}{99}\right)\) | \(e\left(\frac{53}{66}\right)\) |
| \(\chi_{3267}(835,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{173}{198}\right)\) | \(e\left(\frac{74}{99}\right)\) | \(e\left(\frac{32}{99}\right)\) | \(e\left(\frac{23}{198}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{115}{198}\right)\) | \(e\left(\frac{98}{99}\right)\) | \(e\left(\frac{49}{99}\right)\) | \(e\left(\frac{61}{66}\right)\) |
| \(\chi_{3267}(868,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{193}{198}\right)\) | \(e\left(\frac{94}{99}\right)\) | \(e\left(\frac{46}{99}\right)\) | \(e\left(\frac{163}{198}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{23}{198}\right)\) | \(e\left(\frac{79}{99}\right)\) | \(e\left(\frac{89}{99}\right)\) | \(e\left(\frac{65}{66}\right)\) |
| \(\chi_{3267}(934,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{198}\right)\) | \(e\left(\frac{35}{99}\right)\) | \(e\left(\frac{74}{99}\right)\) | \(e\left(\frac{47}{198}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{37}{198}\right)\) | \(e\left(\frac{41}{99}\right)\) | \(e\left(\frac{70}{99}\right)\) | \(e\left(\frac{7}{66}\right)\) |
| \(\chi_{3267}(1033,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{95}{198}\right)\) | \(e\left(\frac{95}{99}\right)\) | \(e\left(\frac{17}{99}\right)\) | \(e\left(\frac{71}{198}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{157}{198}\right)\) | \(e\left(\frac{83}{99}\right)\) | \(e\left(\frac{91}{99}\right)\) | \(e\left(\frac{19}{66}\right)\) |
| \(\chi_{3267}(1066,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{115}{198}\right)\) | \(e\left(\frac{16}{99}\right)\) | \(e\left(\frac{31}{99}\right)\) | \(e\left(\frac{13}{198}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{65}{198}\right)\) | \(e\left(\frac{64}{99}\right)\) | \(e\left(\frac{32}{99}\right)\) | \(e\left(\frac{23}{66}\right)\) |
| \(\chi_{3267}(1132,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{155}{198}\right)\) | \(e\left(\frac{56}{99}\right)\) | \(e\left(\frac{59}{99}\right)\) | \(e\left(\frac{95}{198}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{79}{198}\right)\) | \(e\left(\frac{26}{99}\right)\) | \(e\left(\frac{13}{99}\right)\) | \(e\left(\frac{31}{66}\right)\) |
| \(\chi_{3267}(1165,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{175}{198}\right)\) | \(e\left(\frac{76}{99}\right)\) | \(e\left(\frac{73}{99}\right)\) | \(e\left(\frac{37}{198}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{185}{198}\right)\) | \(e\left(\frac{7}{99}\right)\) | \(e\left(\frac{53}{99}\right)\) | \(e\left(\frac{35}{66}\right)\) |
| \(\chi_{3267}(1231,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{198}\right)\) | \(e\left(\frac{17}{99}\right)\) | \(e\left(\frac{2}{99}\right)\) | \(e\left(\frac{119}{198}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{1}{198}\right)\) | \(e\left(\frac{68}{99}\right)\) | \(e\left(\frac{34}{99}\right)\) | \(e\left(\frac{43}{66}\right)\) |
| \(\chi_{3267}(1264,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{198}\right)\) | \(e\left(\frac{37}{99}\right)\) | \(e\left(\frac{16}{99}\right)\) | \(e\left(\frac{61}{198}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{107}{198}\right)\) | \(e\left(\frac{49}{99}\right)\) | \(e\left(\frac{74}{99}\right)\) | \(e\left(\frac{47}{66}\right)\) |
| \(\chi_{3267}(1363,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{198}\right)\) | \(e\left(\frac{97}{99}\right)\) | \(e\left(\frac{58}{99}\right)\) | \(e\left(\frac{85}{198}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{29}{198}\right)\) | \(e\left(\frac{91}{99}\right)\) | \(e\left(\frac{95}{99}\right)\) | \(e\left(\frac{59}{66}\right)\) |
| \(\chi_{3267}(1429,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{137}{198}\right)\) | \(e\left(\frac{38}{99}\right)\) | \(e\left(\frac{86}{99}\right)\) | \(e\left(\frac{167}{198}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{43}{198}\right)\) | \(e\left(\frac{53}{99}\right)\) | \(e\left(\frac{76}{99}\right)\) | \(e\left(\frac{1}{66}\right)\) |
| \(\chi_{3267}(1462,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{157}{198}\right)\) | \(e\left(\frac{58}{99}\right)\) | \(e\left(\frac{1}{99}\right)\) | \(e\left(\frac{109}{198}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{149}{198}\right)\) | \(e\left(\frac{34}{99}\right)\) | \(e\left(\frac{17}{99}\right)\) | \(e\left(\frac{5}{66}\right)\) |
| \(\chi_{3267}(1528,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{197}{198}\right)\) | \(e\left(\frac{98}{99}\right)\) | \(e\left(\frac{29}{99}\right)\) | \(e\left(\frac{191}{198}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{163}{198}\right)\) | \(e\left(\frac{95}{99}\right)\) | \(e\left(\frac{97}{99}\right)\) | \(e\left(\frac{13}{66}\right)\) |
| \(\chi_{3267}(1561,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{198}\right)\) | \(e\left(\frac{19}{99}\right)\) | \(e\left(\frac{43}{99}\right)\) | \(e\left(\frac{133}{198}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{71}{198}\right)\) | \(e\left(\frac{76}{99}\right)\) | \(e\left(\frac{38}{99}\right)\) | \(e\left(\frac{17}{66}\right)\) |
| \(\chi_{3267}(1627,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{198}\right)\) | \(e\left(\frac{59}{99}\right)\) | \(e\left(\frac{71}{99}\right)\) | \(e\left(\frac{17}{198}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{85}{198}\right)\) | \(e\left(\frac{38}{99}\right)\) | \(e\left(\frac{19}{99}\right)\) | \(e\left(\frac{25}{66}\right)\) |
| \(\chi_{3267}(1660,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{198}\right)\) | \(e\left(\frac{79}{99}\right)\) | \(e\left(\frac{85}{99}\right)\) | \(e\left(\frac{157}{198}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{191}{198}\right)\) | \(e\left(\frac{19}{99}\right)\) | \(e\left(\frac{59}{99}\right)\) | \(e\left(\frac{29}{66}\right)\) |