from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6422, base_ring=CyclotomicField(234))
M = H._module
chi = DirichletCharacter(H, M([171,65]))
chi.galois_orbit()
[g,chi] = znchar(Mod(51,6422))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(6422\) | |
| Conductor: | \(3211\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(234\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 3211.dh | 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: | $\Q(\zeta_{117})$ |
| Fixed field: | Number field defined by a degree 234 polynomial (not computed) |
First 31 of 72 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{6422}(51,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{234}\right)\) | \(e\left(\frac{5}{234}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{53}{117}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{29}{117}\right)\) | \(e\left(\frac{55}{117}\right)\) | \(e\left(\frac{10}{117}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{117}\right)\) |
| \(\chi_{6422}(129,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{234}\right)\) | \(e\left(\frac{227}{234}\right)\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{113}{117}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{53}{117}\right)\) | \(e\left(\frac{40}{117}\right)\) | \(e\left(\frac{103}{117}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{110}{117}\right)\) |
| \(\chi_{6422}(155,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{234}\right)\) | \(e\left(\frac{67}{234}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{55}{117}\right)\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{61}{117}\right)\) | \(e\left(\frac{35}{117}\right)\) | \(e\left(\frac{17}{117}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{67}{117}\right)\) |
| \(\chi_{6422}(181,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{234}\right)\) | \(e\left(\frac{89}{234}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{101}{117}\right)\) | \(e\left(\frac{41}{78}\right)\) | \(e\left(\frac{95}{117}\right)\) | \(e\left(\frac{43}{117}\right)\) | \(e\left(\frac{61}{117}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{89}{117}\right)\) |
| \(\chi_{6422}(363,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{234}\right)\) | \(e\left(\frac{217}{234}\right)\) | \(e\left(\frac{53}{78}\right)\) | \(e\left(\frac{7}{117}\right)\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{112}{117}\right)\) | \(e\left(\frac{47}{117}\right)\) | \(e\left(\frac{83}{117}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{100}{117}\right)\) |
| \(\chi_{6422}(545,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{234}\right)\) | \(e\left(\frac{59}{234}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{17}{117}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{38}{117}\right)\) | \(e\left(\frac{64}{117}\right)\) | \(e\left(\frac{1}{117}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{59}{117}\right)\) |
| \(\chi_{6422}(623,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{234}\right)\) | \(e\left(\frac{47}{234}\right)\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{77}{117}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{62}{117}\right)\) | \(e\left(\frac{49}{117}\right)\) | \(e\left(\frac{94}{117}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{47}{117}\right)\) |
| \(\chi_{6422}(649,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{234}\right)\) | \(e\left(\frac{121}{234}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{19}{117}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{70}{117}\right)\) | \(e\left(\frac{44}{117}\right)\) | \(e\left(\frac{8}{117}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{4}{117}\right)\) |
| \(\chi_{6422}(831,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{211}{234}\right)\) | \(e\left(\frac{223}{234}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{94}{117}\right)\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{100}{117}\right)\) | \(e\left(\frac{113}{117}\right)\) | \(e\left(\frac{95}{117}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{106}{117}\right)\) |
| \(\chi_{6422}(857,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{205}{234}\right)\) | \(e\left(\frac{37}{234}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{88}{117}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{4}{117}\right)\) | \(e\left(\frac{56}{117}\right)\) | \(e\left(\frac{74}{117}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{37}{117}\right)\) |
| \(\chi_{6422}(1039,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{215}{234}\right)\) | \(e\left(\frac{113}{234}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{98}{117}\right)\) | \(e\left(\frac{17}{78}\right)\) | \(e\left(\frac{47}{117}\right)\) | \(e\left(\frac{73}{117}\right)\) | \(e\left(\frac{109}{117}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{113}{117}\right)\) |
| \(\chi_{6422}(1117,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{234}\right)\) | \(e\left(\frac{101}{234}\right)\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{41}{117}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{71}{117}\right)\) | \(e\left(\frac{58}{117}\right)\) | \(e\left(\frac{85}{117}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{101}{117}\right)\) |
| \(\chi_{6422}(1143,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{217}{234}\right)\) | \(e\left(\frac{175}{234}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{100}{117}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{79}{117}\right)\) | \(e\left(\frac{53}{117}\right)\) | \(e\left(\frac{116}{117}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{58}{117}\right)\) |
| \(\chi_{6422}(1169,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{234}\right)\) | \(e\left(\frac{197}{234}\right)\) | \(e\left(\frac{19}{78}\right)\) | \(e\left(\frac{29}{117}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{113}{117}\right)\) | \(e\left(\frac{61}{117}\right)\) | \(e\left(\frac{43}{117}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{80}{117}\right)\) |
| \(\chi_{6422}(1325,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{175}{234}\right)\) | \(e\left(\frac{43}{234}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{58}{117}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{109}{117}\right)\) | \(e\left(\frac{5}{117}\right)\) | \(e\left(\frac{86}{117}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{43}{117}\right)\) |
| \(\chi_{6422}(1533,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{179}{234}\right)\) | \(e\left(\frac{167}{234}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{62}{117}\right)\) | \(e\left(\frac{41}{78}\right)\) | \(e\left(\frac{56}{117}\right)\) | \(e\left(\frac{82}{117}\right)\) | \(e\left(\frac{100}{117}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{50}{117}\right)\) |
| \(\chi_{6422}(1611,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{234}\right)\) | \(e\left(\frac{155}{234}\right)\) | \(e\left(\frac{49}{78}\right)\) | \(e\left(\frac{5}{117}\right)\) | \(e\left(\frac{53}{78}\right)\) | \(e\left(\frac{80}{117}\right)\) | \(e\left(\frac{67}{117}\right)\) | \(e\left(\frac{76}{117}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{38}{117}\right)\) |
| \(\chi_{6422}(1637,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{181}{234}\right)\) | \(e\left(\frac{229}{234}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{64}{117}\right)\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{88}{117}\right)\) | \(e\left(\frac{62}{117}\right)\) | \(e\left(\frac{107}{117}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{112}{117}\right)\) |
| \(\chi_{6422}(1663,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{227}{234}\right)\) | \(e\left(\frac{17}{234}\right)\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{110}{117}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{5}{117}\right)\) | \(e\left(\frac{70}{117}\right)\) | \(e\left(\frac{34}{117}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{17}{117}\right)\) |
| \(\chi_{6422}(1819,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{139}{234}\right)\) | \(e\left(\frac{97}{234}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{22}{117}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{1}{117}\right)\) | \(e\left(\frac{14}{117}\right)\) | \(e\left(\frac{77}{117}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{97}{117}\right)\) |
| \(\chi_{6422}(1845,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{133}{234}\right)\) | \(e\left(\frac{145}{234}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{16}{117}\right)\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{22}{117}\right)\) | \(e\left(\frac{74}{117}\right)\) | \(e\left(\frac{56}{117}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{28}{117}\right)\) |
| \(\chi_{6422}(2105,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{203}{234}\right)\) | \(e\left(\frac{209}{234}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{86}{117}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{89}{117}\right)\) | \(e\left(\frac{76}{117}\right)\) | \(e\left(\frac{67}{117}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{92}{117}\right)\) |
| \(\chi_{6422}(2131,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{145}{234}\right)\) | \(e\left(\frac{49}{234}\right)\) | \(e\left(\frac{17}{78}\right)\) | \(e\left(\frac{28}{117}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{97}{117}\right)\) | \(e\left(\frac{71}{117}\right)\) | \(e\left(\frac{98}{117}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{49}{117}\right)\) |
| \(\chi_{6422}(2157,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{191}{234}\right)\) | \(e\left(\frac{71}{234}\right)\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{74}{117}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{14}{117}\right)\) | \(e\left(\frac{79}{117}\right)\) | \(e\left(\frac{25}{117}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{71}{117}\right)\) |
| \(\chi_{6422}(2313,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{103}{234}\right)\) | \(e\left(\frac{151}{234}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{103}{117}\right)\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{10}{117}\right)\) | \(e\left(\frac{23}{117}\right)\) | \(e\left(\frac{68}{117}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{34}{117}\right)\) |
| \(\chi_{6422}(2339,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{234}\right)\) | \(e\left(\frac{199}{234}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{97}{117}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{31}{117}\right)\) | \(e\left(\frac{83}{117}\right)\) | \(e\left(\frac{47}{117}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{82}{117}\right)\) |
| \(\chi_{6422}(2521,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{234}\right)\) | \(e\left(\frac{41}{234}\right)\) | \(e\left(\frac{19}{78}\right)\) | \(e\left(\frac{107}{117}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{74}{117}\right)\) | \(e\left(\frac{100}{117}\right)\) | \(e\left(\frac{82}{117}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{41}{117}\right)\) |
| \(\chi_{6422}(2599,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{167}{234}\right)\) | \(e\left(\frac{29}{234}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{50}{117}\right)\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{98}{117}\right)\) | \(e\left(\frac{85}{117}\right)\) | \(e\left(\frac{58}{117}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{29}{117}\right)\) |
| \(\chi_{6422}(2625,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{234}\right)\) | \(e\left(\frac{103}{234}\right)\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{109}{117}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{106}{117}\right)\) | \(e\left(\frac{80}{117}\right)\) | \(e\left(\frac{89}{117}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{103}{117}\right)\) |
| \(\chi_{6422}(2651,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{155}{234}\right)\) | \(e\left(\frac{125}{234}\right)\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{38}{117}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{23}{117}\right)\) | \(e\left(\frac{88}{117}\right)\) | \(e\left(\frac{16}{117}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{8}{117}\right)\) |
| \(\chi_{6422}(2807,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{234}\right)\) | \(e\left(\frac{205}{234}\right)\) | \(e\left(\frac{17}{78}\right)\) | \(e\left(\frac{67}{117}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{19}{117}\right)\) | \(e\left(\frac{32}{117}\right)\) | \(e\left(\frac{59}{117}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{88}{117}\right)\) |