from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8001, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([105,63,80]))
chi.galois_orbit()
[g,chi] = znchar(Mod(41,8001))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8001\) | |
| Conductor: | \(8001\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(126\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{63})$ |
| Fixed field: | Number field defined by a degree 126 polynomial (not computed) |
First 31 of 36 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8001}(41,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{1}{126}\right)\) | \(e\left(\frac{107}{126}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{8001}(335,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{41}{126}\right)\) | \(e\left(\frac{103}{126}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{8001}(587,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{17}{126}\right)\) | \(e\left(\frac{55}{126}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{8001}(650,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{83}{126}\right)\) | \(e\left(\frac{61}{126}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{8001}(671,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{79}{126}\right)\) | \(e\left(\frac{11}{126}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{8001}(860,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{103}{126}\right)\) | \(e\left(\frac{59}{126}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{8001}(902,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{113}{126}\right)\) | \(e\left(\frac{121}{126}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{8001}(923,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{25}{126}\right)\) | \(e\left(\frac{29}{126}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{8001}(1595,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{101}{126}\right)\) | \(e\left(\frac{97}{126}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{8001}(1721,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{5}{126}\right)\) | \(e\left(\frac{31}{126}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{40}{63}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{8001}(2120,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{13}{126}\right)\) | \(e\left(\frac{5}{126}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{41}{63}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{8001}(2561,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{55}{126}\right)\) | \(e\left(\frac{89}{126}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{8001}(2624,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{19}{126}\right)\) | \(e\left(\frac{17}{126}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{8001}(2729,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{107}{126}\right)\) | \(e\left(\frac{109}{126}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{8001}(2876,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{109}{126}\right)\) | \(e\left(\frac{71}{126}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{8001}(2918,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{89}{126}\right)\) | \(e\left(\frac{73}{126}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{8001}(2939,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{97}{126}\right)\) | \(e\left(\frac{47}{126}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{20}{63}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{8001}(2981,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{47}{126}\right)\) | \(e\left(\frac{115}{126}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{8001}(3065,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{43}{126}\right)\) | \(e\left(\frac{65}{126}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{8001}(3296,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{71}{126}\right)\) | \(e\left(\frac{37}{126}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{8001}(3422,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{29}{126}\right)\) | \(e\left(\frac{79}{126}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{8001}(3821,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{67}{126}\right)\) | \(e\left(\frac{113}{126}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{8001}(3884,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{73}{126}\right)\) | \(e\left(\frac{125}{126}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{8001}(4052,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{53}{126}\right)\) | \(e\left(\frac{1}{126}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{8001}(4073,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{115}{126}\right)\) | \(e\left(\frac{83}{126}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{8001}(4136,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{61}{126}\right)\) | \(e\left(\frac{101}{126}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{8001}(4304,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{11}{126}\right)\) | \(e\left(\frac{43}{126}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{25}{63}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{8001}(4514,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{85}{126}\right)\) | \(e\left(\frac{23}{126}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{8001}(5438,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{59}{126}\right)\) | \(e\left(\frac{13}{126}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{8001}(6131,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{23}{126}\right)\) | \(e\left(\frac{67}{126}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{8001}(6635,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{125}{126}\right)\) | \(e\left(\frac{19}{126}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{1}{6}\right)\) |