from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(199, base_ring=CyclotomicField(198))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,199))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(199\) | |
| Conductor: | \(199\) | 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{199}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{99}\right)\) | \(e\left(\frac{1}{198}\right)\) | \(e\left(\frac{7}{99}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{107}{198}\right)\) | \(e\left(\frac{71}{99}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{1}{99}\right)\) | \(e\left(\frac{23}{99}\right)\) | \(e\left(\frac{21}{22}\right)\) |
| \(\chi_{199}(6,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{28}{99}\right)\) | \(e\left(\frac{107}{198}\right)\) | \(e\left(\frac{56}{99}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{163}{198}\right)\) | \(e\left(\frac{73}{99}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{8}{99}\right)\) | \(e\left(\frac{85}{99}\right)\) | \(e\left(\frac{3}{22}\right)\) |
| \(\chi_{199}(15,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{99}\right)\) | \(e\left(\frac{139}{198}\right)\) | \(e\left(\frac{82}{99}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{23}{198}\right)\) | \(e\left(\frac{68}{99}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{40}{99}\right)\) | \(e\left(\frac{29}{99}\right)\) | \(e\left(\frac{15}{22}\right)\) |
| \(\chi_{199}(22,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{92}{99}\right)\) | \(e\left(\frac{97}{198}\right)\) | \(e\left(\frac{85}{99}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{83}{198}\right)\) | \(e\left(\frac{56}{99}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{97}{99}\right)\) | \(e\left(\frac{53}{99}\right)\) | \(e\left(\frac{13}{22}\right)\) |
| \(\chi_{199}(30,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{16}{99}\right)\) | \(e\left(\frac{47}{198}\right)\) | \(e\left(\frac{32}{99}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{79}{198}\right)\) | \(e\left(\frac{70}{99}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{47}{99}\right)\) | \(e\left(\frac{91}{99}\right)\) | \(e\left(\frac{19}{22}\right)\) |
| \(\chi_{199}(34,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{99}\right)\) | \(e\left(\frac{31}{198}\right)\) | \(e\left(\frac{19}{99}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{149}{198}\right)\) | \(e\left(\frac{23}{99}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{31}{99}\right)\) | \(e\left(\frac{20}{99}\right)\) | \(e\left(\frac{13}{22}\right)\) |
| \(\chi_{199}(38,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{99}\right)\) | \(e\left(\frac{161}{198}\right)\) | \(e\left(\frac{38}{99}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{1}{198}\right)\) | \(e\left(\frac{46}{99}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{62}{99}\right)\) | \(e\left(\frac{40}{99}\right)\) | \(e\left(\frac{15}{22}\right)\) |
| \(\chi_{199}(39,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{99}\right)\) | \(e\left(\frac{173}{198}\right)\) | \(e\left(\frac{23}{99}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{97}{198}\right)\) | \(e\left(\frac{7}{99}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{74}{99}\right)\) | \(e\left(\frac{19}{99}\right)\) | \(e\left(\frac{3}{22}\right)\) |
| \(\chi_{199}(41,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{40}{99}\right)\) | \(e\left(\frac{167}{198}\right)\) | \(e\left(\frac{80}{99}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{49}{198}\right)\) | \(e\left(\frac{76}{99}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{68}{99}\right)\) | \(e\left(\frac{79}{99}\right)\) | \(e\left(\frac{9}{22}\right)\) |
| \(\chi_{199}(44,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{99}\right)\) | \(e\left(\frac{5}{198}\right)\) | \(e\left(\frac{35}{99}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{139}{198}\right)\) | \(e\left(\frac{58}{99}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{5}{99}\right)\) | \(e\left(\frac{16}{99}\right)\) | \(e\left(\frac{17}{22}\right)\) |
| \(\chi_{199}(48,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{52}{99}\right)\) | \(e\left(\frac{29}{198}\right)\) | \(e\left(\frac{5}{99}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{133}{198}\right)\) | \(e\left(\frac{79}{99}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{29}{99}\right)\) | \(e\left(\frac{73}{99}\right)\) | \(e\left(\frac{15}{22}\right)\) |
| \(\chi_{199}(54,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{99}\right)\) | \(e\left(\frac{109}{198}\right)\) | \(e\left(\frac{70}{99}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{179}{198}\right)\) | \(e\left(\frac{17}{99}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{10}{99}\right)\) | \(e\left(\frac{32}{99}\right)\) | \(e\left(\frac{1}{22}\right)\) |
| \(\chi_{199}(68,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{34}{99}\right)\) | \(e\left(\frac{137}{198}\right)\) | \(e\left(\frac{68}{99}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{7}{198}\right)\) | \(e\left(\frac{25}{99}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{38}{99}\right)\) | \(e\left(\frac{82}{99}\right)\) | \(e\left(\frac{17}{22}\right)\) |
| \(\chi_{199}(69,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{70}{99}\right)\) | \(e\left(\frac{119}{198}\right)\) | \(e\left(\frac{41}{99}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{61}{198}\right)\) | \(e\left(\frac{34}{99}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{20}{99}\right)\) | \(e\left(\frac{64}{99}\right)\) | \(e\left(\frac{13}{22}\right)\) |
| \(\chi_{199}(71,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{99}\right)\) | \(e\left(\frac{49}{198}\right)\) | \(e\left(\frac{46}{99}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{95}{198}\right)\) | \(e\left(\frac{14}{99}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{49}{99}\right)\) | \(e\left(\frac{38}{99}\right)\) | \(e\left(\frac{17}{22}\right)\) |
| \(\chi_{199}(73,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{99}\right)\) | \(e\left(\frac{151}{198}\right)\) | \(e\left(\frac{67}{99}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{119}{198}\right)\) | \(e\left(\frac{29}{99}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{52}{99}\right)\) | \(e\left(\frac{8}{99}\right)\) | \(e\left(\frac{3}{22}\right)\) |
| \(\chi_{199}(75,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{99}\right)\) | \(e\left(\frac{79}{198}\right)\) | \(e\left(\frac{58}{99}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{137}{198}\right)\) | \(e\left(\frac{65}{99}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{79}{99}\right)\) | \(e\left(\frac{35}{99}\right)\) | \(e\left(\frac{9}{22}\right)\) |
| \(\chi_{199}(77,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{20}{99}\right)\) | \(e\left(\frac{133}{198}\right)\) | \(e\left(\frac{40}{99}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{173}{198}\right)\) | \(e\left(\frac{38}{99}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{34}{99}\right)\) | \(e\left(\frac{89}{99}\right)\) | \(e\left(\frac{21}{22}\right)\) |
| \(\chi_{199}(84,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{99}\right)\) | \(e\left(\frac{157}{198}\right)\) | \(e\left(\frac{10}{99}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{167}{198}\right)\) | \(e\left(\frac{59}{99}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{58}{99}\right)\) | \(e\left(\frac{47}{99}\right)\) | \(e\left(\frac{19}{22}\right)\) |
| \(\chi_{199}(87,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{99}\right)\) | \(e\left(\frac{71}{198}\right)\) | \(e\left(\frac{2}{99}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{73}{198}\right)\) | \(e\left(\frac{91}{99}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{71}{99}\right)\) | \(e\left(\frac{49}{99}\right)\) | \(e\left(\frac{17}{22}\right)\) |
| \(\chi_{199}(95,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{32}{99}\right)\) | \(e\left(\frac{193}{198}\right)\) | \(e\left(\frac{64}{99}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{59}{198}\right)\) | \(e\left(\frac{41}{99}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{94}{99}\right)\) | \(e\left(\frac{83}{99}\right)\) | \(e\left(\frac{5}{22}\right)\) |
| \(\chi_{199}(97,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{99}\right)\) | \(e\left(\frac{131}{198}\right)\) | \(e\left(\frac{26}{99}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{157}{198}\right)\) | \(e\left(\frac{94}{99}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{32}{99}\right)\) | \(e\left(\frac{43}{99}\right)\) | \(e\left(\frac{1}{22}\right)\) |
| \(\chi_{199}(99,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{99}\right)\) | \(e\left(\frac{191}{198}\right)\) | \(e\left(\frac{50}{99}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{43}{198}\right)\) | \(e\left(\frac{97}{99}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{92}{99}\right)\) | \(e\left(\frac{37}{99}\right)\) | \(e\left(\frac{7}{22}\right)\) |
| \(\chi_{199}(105,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{99}\right)\) | \(e\left(\frac{83}{198}\right)\) | \(e\left(\frac{86}{99}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{169}{198}\right)\) | \(e\left(\frac{52}{99}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{83}{99}\right)\) | \(e\left(\frac{28}{99}\right)\) | \(e\left(\frac{5}{22}\right)\) |
| \(\chi_{199}(108,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{10}{99}\right)\) | \(e\left(\frac{17}{198}\right)\) | \(e\left(\frac{20}{99}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{37}{198}\right)\) | \(e\left(\frac{19}{99}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{17}{99}\right)\) | \(e\left(\frac{94}{99}\right)\) | \(e\left(\frac{5}{22}\right)\) |
| \(\chi_{199}(110,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{80}{99}\right)\) | \(e\left(\frac{37}{198}\right)\) | \(e\left(\frac{61}{99}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{197}{198}\right)\) | \(e\left(\frac{53}{99}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{37}{99}\right)\) | \(e\left(\frac{59}{99}\right)\) | \(e\left(\frac{7}{22}\right)\) |
| \(\chi_{199}(113,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{85}{99}\right)\) | \(e\left(\frac{95}{198}\right)\) | \(e\left(\frac{71}{99}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{67}{198}\right)\) | \(e\left(\frac{13}{99}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{95}{99}\right)\) | \(e\left(\frac{7}{99}\right)\) | \(e\left(\frac{15}{22}\right)\) |
| \(\chi_{199}(118,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{14}{99}\right)\) | \(e\left(\frac{103}{198}\right)\) | \(e\left(\frac{28}{99}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{131}{198}\right)\) | \(e\left(\frac{86}{99}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{4}{99}\right)\) | \(e\left(\frac{92}{99}\right)\) | \(e\left(\frac{7}{22}\right)\) |
| \(\chi_{199}(119,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{86}{99}\right)\) | \(e\left(\frac{67}{198}\right)\) | \(e\left(\frac{73}{99}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{41}{198}\right)\) | \(e\left(\frac{5}{99}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{67}{99}\right)\) | \(e\left(\frac{56}{99}\right)\) | \(e\left(\frac{21}{22}\right)\) |
| \(\chi_{199}(120,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{99}\right)\) | \(e\left(\frac{61}{198}\right)\) | \(e\left(\frac{31}{99}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{191}{198}\right)\) | \(e\left(\frac{74}{99}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{61}{99}\right)\) | \(e\left(\frac{17}{99}\right)\) | \(e\left(\frac{5}{22}\right)\) |
| \(\chi_{199}(127,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{99}\right)\) | \(e\left(\frac{23}{198}\right)\) | \(e\left(\frac{62}{99}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{85}{198}\right)\) | \(e\left(\frac{49}{99}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{23}{99}\right)\) | \(e\left(\frac{34}{99}\right)\) | \(e\left(\frac{21}{22}\right)\) |