from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(597, base_ring=CyclotomicField(198))
M = H._module
chi = DirichletCharacter(H, M([0,97]))
chi.galois_orbit()
[g,chi] = znchar(Mod(22,597))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(597\) | |
| Conductor: | \(199\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(198\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 199.l | 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 | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{597}(22,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{92}{99}\right)\) | \(e\left(\frac{85}{99}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{56}{99}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{53}{99}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{26}{99}\right)\) | \(e\left(\frac{49}{99}\right)\) | \(e\left(\frac{71}{99}\right)\) |
| \(\chi_{597}(34,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{99}\right)\) | \(e\left(\frac{19}{99}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{23}{99}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{20}{99}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{92}{99}\right)\) | \(e\left(\frac{82}{99}\right)\) | \(e\left(\frac{38}{99}\right)\) |
| \(\chi_{597}(73,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{99}\right)\) | \(e\left(\frac{67}{99}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{29}{99}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{8}{99}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{17}{99}\right)\) | \(e\left(\frac{13}{99}\right)\) | \(e\left(\frac{35}{99}\right)\) |
| \(\chi_{597}(97,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{99}\right)\) | \(e\left(\frac{26}{99}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{94}{99}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{43}{99}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{79}{99}\right)\) | \(e\left(\frac{8}{99}\right)\) | \(e\left(\frac{52}{99}\right)\) |
| \(\chi_{597}(118,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{14}{99}\right)\) | \(e\left(\frac{28}{99}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{86}{99}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{92}{99}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{47}{99}\right)\) | \(e\left(\frac{1}{99}\right)\) | \(e\left(\frac{56}{99}\right)\) |
| \(\chi_{597}(127,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{99}\right)\) | \(e\left(\frac{62}{99}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{49}{99}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{34}{99}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{97}{99}\right)\) | \(e\left(\frac{80}{99}\right)\) | \(e\left(\frac{25}{99}\right)\) |
| \(\chi_{597}(133,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{46}{99}\right)\) | \(e\left(\frac{92}{99}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{28}{99}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{76}{99}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{13}{99}\right)\) | \(e\left(\frac{74}{99}\right)\) | \(e\left(\frac{85}{99}\right)\) |
| \(\chi_{597}(142,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{99}\right)\) | \(e\left(\frac{95}{99}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{16}{99}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{1}{99}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{64}{99}\right)\) | \(e\left(\frac{14}{99}\right)\) | \(e\left(\frac{91}{99}\right)\) |
| \(\chi_{597}(148,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{38}{99}\right)\) | \(e\left(\frac{76}{99}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{92}{99}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{80}{99}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{71}{99}\right)\) | \(e\left(\frac{31}{99}\right)\) | \(e\left(\frac{53}{99}\right)\) |
| \(\chi_{597}(154,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{94}{99}\right)\) | \(e\left(\frac{89}{99}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{40}{99}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{52}{99}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{61}{99}\right)\) | \(e\left(\frac{35}{99}\right)\) | \(e\left(\frac{79}{99}\right)\) |
| \(\chi_{597}(163,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{56}{99}\right)\) | \(e\left(\frac{13}{99}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{47}{99}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{71}{99}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{89}{99}\right)\) | \(e\left(\frac{4}{99}\right)\) | \(e\left(\frac{26}{99}\right)\) |
| \(\chi_{597}(166,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{99}\right)\) | \(e\left(\frac{43}{99}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{26}{99}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{14}{99}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{5}{99}\right)\) | \(e\left(\frac{97}{99}\right)\) | \(e\left(\frac{86}{99}\right)\) |
| \(\chi_{597}(190,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{99}\right)\) | \(e\left(\frac{14}{99}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{43}{99}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{46}{99}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{73}{99}\right)\) | \(e\left(\frac{50}{99}\right)\) | \(e\left(\frac{28}{99}\right)\) |
| \(\chi_{597}(202,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{99}\right)\) | \(e\left(\frac{7}{99}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{71}{99}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{23}{99}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{86}{99}\right)\) | \(e\left(\frac{25}{99}\right)\) | \(e\left(\frac{14}{99}\right)\) |
| \(\chi_{597}(205,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{28}{99}\right)\) | \(e\left(\frac{56}{99}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{73}{99}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{85}{99}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{94}{99}\right)\) | \(e\left(\frac{2}{99}\right)\) | \(e\left(\frac{13}{99}\right)\) |
| \(\chi_{597}(214,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{99}\right)\) | \(e\left(\frac{82}{99}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{68}{99}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{29}{99}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{74}{99}\right)\) | \(e\left(\frac{10}{99}\right)\) | \(e\left(\frac{65}{99}\right)\) |
| \(\chi_{597}(229,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{16}{99}\right)\) | \(e\left(\frac{32}{99}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{70}{99}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{91}{99}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{82}{99}\right)\) | \(e\left(\frac{86}{99}\right)\) | \(e\left(\frac{64}{99}\right)\) |
| \(\chi_{597}(238,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{99}\right)\) | \(e\left(\frac{23}{99}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{7}{99}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{19}{99}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{28}{99}\right)\) | \(e\left(\frac{68}{99}\right)\) | \(e\left(\frac{46}{99}\right)\) |
| \(\chi_{597}(247,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{52}{99}\right)\) | \(e\left(\frac{5}{99}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{79}{99}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{73}{99}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{19}{99}\right)\) | \(e\left(\frac{32}{99}\right)\) | \(e\left(\frac{10}{99}\right)\) |
| \(\chi_{597}(253,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{99}\right)\) | \(e\left(\frac{70}{99}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{17}{99}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{32}{99}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{68}{99}\right)\) | \(e\left(\frac{52}{99}\right)\) | \(e\left(\frac{41}{99}\right)\) |
| \(\chi_{597}(268,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{70}{99}\right)\) | \(e\left(\frac{41}{99}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{34}{99}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{64}{99}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{37}{99}\right)\) | \(e\left(\frac{5}{99}\right)\) | \(e\left(\frac{82}{99}\right)\) |
| \(\chi_{597}(274,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{99}\right)\) | \(e\left(\frac{58}{99}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{65}{99}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{35}{99}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{62}{99}\right)\) | \(e\left(\frac{94}{99}\right)\) | \(e\left(\frac{17}{99}\right)\) |
| \(\chi_{597}(283,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{99}\right)\) | \(e\left(\frac{10}{99}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{59}{99}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{47}{99}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{38}{99}\right)\) | \(e\left(\frac{64}{99}\right)\) | \(e\left(\frac{20}{99}\right)\) |
| \(\chi_{597}(286,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{99}\right)\) | \(e\left(\frac{2}{99}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{91}{99}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{49}{99}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{67}{99}\right)\) | \(e\left(\frac{92}{99}\right)\) | \(e\left(\frac{4}{99}\right)\) |
| \(\chi_{597}(298,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{99}\right)\) | \(e\left(\frac{50}{99}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{97}{99}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{37}{99}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{91}{99}\right)\) | \(e\left(\frac{23}{99}\right)\) | \(e\left(\frac{1}{99}\right)\) |
| \(\chi_{597}(304,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{99}\right)\) | \(e\left(\frac{86}{99}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{52}{99}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{28}{99}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{10}{99}\right)\) | \(e\left(\frac{95}{99}\right)\) | \(e\left(\frac{73}{99}\right)\) |
| \(\chi_{597}(307,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{10}{99}\right)\) | \(e\left(\frac{20}{99}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{19}{99}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{94}{99}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{76}{99}\right)\) | \(e\left(\frac{29}{99}\right)\) | \(e\left(\frac{40}{99}\right)\) |
| \(\chi_{597}(319,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{99}\right)\) | \(e\left(\frac{31}{99}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{74}{99}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{17}{99}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{98}{99}\right)\) | \(e\left(\frac{40}{99}\right)\) | \(e\left(\frac{62}{99}\right)\) |
| \(\chi_{597}(328,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{64}{99}\right)\) | \(e\left(\frac{29}{99}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{82}{99}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{67}{99}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{31}{99}\right)\) | \(e\left(\frac{47}{99}\right)\) | \(e\left(\frac{58}{99}\right)\) |
| \(\chi_{597}(349,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{99}\right)\) | \(e\left(\frac{8}{99}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{67}{99}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{97}{99}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{70}{99}\right)\) | \(e\left(\frac{71}{99}\right)\) | \(e\left(\frac{16}{99}\right)\) |
| \(\chi_{597}(352,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{99}\right)\) | \(e\left(\frac{83}{99}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{64}{99}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{4}{99}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{58}{99}\right)\) | \(e\left(\frac{56}{99}\right)\) | \(e\left(\frac{67}{99}\right)\) |