from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3724, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([63,3,91]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,3724))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3724\) | |
| Conductor: | \(3724\) | 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: | odd | 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 | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(23\) | \(25\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3724}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{115}{126}\right)\) | \(e\left(\frac{31}{126}\right)\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{25}{63}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{103}{126}\right)\) | \(e\left(\frac{107}{126}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{31}{42}\right)\) |
| \(\chi_{3724}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{126}\right)\) | \(e\left(\frac{109}{126}\right)\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{25}{63}\right)\) | \(e\left(\frac{37}{126}\right)\) | \(e\left(\frac{47}{126}\right)\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{25}{42}\right)\) |
| \(\chi_{3724}(243,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{126}\right)\) | \(e\left(\frac{29}{126}\right)\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{11}{126}\right)\) | \(e\left(\frac{31}{126}\right)\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{29}{42}\right)\) |
| \(\chi_{3724}(355,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{126}\right)\) | \(e\left(\frac{5}{126}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{41}{126}\right)\) | \(e\left(\frac{1}{126}\right)\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{5}{42}\right)\) |
| \(\chi_{3724}(451,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{126}\right)\) | \(e\left(\frac{25}{126}\right)\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{79}{126}\right)\) | \(e\left(\frac{5}{126}\right)\) | \(e\left(\frac{25}{63}\right)\) | \(e\left(\frac{25}{42}\right)\) |
| \(\chi_{3724}(523,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{95}{126}\right)\) | \(e\left(\frac{53}{126}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{107}{126}\right)\) | \(e\left(\frac{61}{126}\right)\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{11}{42}\right)\) |
| \(\chi_{3724}(535,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{126}\right)\) | \(e\left(\frac{121}{126}\right)\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{85}{126}\right)\) | \(e\left(\frac{125}{126}\right)\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{37}{42}\right)\) |
| \(\chi_{3724}(591,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{126}\right)\) | \(e\left(\frac{73}{126}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{19}{126}\right)\) | \(e\left(\frac{65}{126}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{31}{42}\right)\) |
| \(\chi_{3724}(775,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{125}{126}\right)\) | \(e\left(\frac{83}{126}\right)\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{41}{63}\right)\) | \(e\left(\frac{101}{126}\right)\) | \(e\left(\frac{67}{126}\right)\) | \(e\left(\frac{20}{63}\right)\) | \(e\left(\frac{41}{42}\right)\) |
| \(\chi_{3724}(887,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{126}\right)\) | \(e\left(\frac{59}{126}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{5}{126}\right)\) | \(e\left(\frac{37}{126}\right)\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{17}{42}\right)\) |
| \(\chi_{3724}(983,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{126}\right)\) | \(e\left(\frac{115}{126}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{61}{126}\right)\) | \(e\left(\frac{23}{126}\right)\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{31}{42}\right)\) |
| \(\chi_{3724}(1055,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{126}\right)\) | \(e\left(\frac{107}{126}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{71}{126}\right)\) | \(e\left(\frac{97}{126}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{23}{42}\right)\) |
| \(\chi_{3724}(1067,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{126}\right)\) | \(e\left(\frac{85}{126}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{67}{126}\right)\) | \(e\left(\frac{17}{126}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{1}{42}\right)\) |
| \(\chi_{3724}(1123,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{121}{126}\right)\) | \(e\left(\frac{37}{126}\right)\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{40}{63}\right)\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{1}{126}\right)\) | \(e\left(\frac{83}{126}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{37}{42}\right)\) |
| \(\chi_{3724}(1307,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{126}\right)\) | \(e\left(\frac{11}{126}\right)\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{65}{126}\right)\) | \(e\left(\frac{103}{126}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{11}{42}\right)\) |
| \(\chi_{3724}(1419,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{126}\right)\) | \(e\left(\frac{113}{126}\right)\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{20}{63}\right)\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{95}{126}\right)\) | \(e\left(\frac{73}{126}\right)\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{29}{42}\right)\) |
| \(\chi_{3724}(1515,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{126}\right)\) | \(e\left(\frac{79}{126}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{43}{126}\right)\) | \(e\left(\frac{41}{126}\right)\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{37}{42}\right)\) |
| \(\chi_{3724}(1655,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{85}{126}\right)\) | \(e\left(\frac{1}{126}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{109}{126}\right)\) | \(e\left(\frac{101}{126}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{1}{42}\right)\) |
| \(\chi_{3724}(1839,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{126}\right)\) | \(e\left(\frac{65}{126}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{29}{126}\right)\) | \(e\left(\frac{13}{126}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{23}{42}\right)\) |
| \(\chi_{3724}(1951,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{126}\right)\) | \(e\left(\frac{41}{126}\right)\) | \(e\left(\frac{20}{63}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{59}{126}\right)\) | \(e\left(\frac{109}{126}\right)\) | \(e\left(\frac{41}{63}\right)\) | \(e\left(\frac{41}{42}\right)\) |
| \(\chi_{3724}(2047,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{126}\right)\) | \(e\left(\frac{43}{126}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{25}{126}\right)\) | \(e\left(\frac{59}{126}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{1}{42}\right)\) |
| \(\chi_{3724}(2119,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{126}\right)\) | \(e\left(\frac{89}{126}\right)\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{125}{126}\right)\) | \(e\left(\frac{43}{126}\right)\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{5}{42}\right)\) |
| \(\chi_{3724}(2131,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{126}\right)\) | \(e\left(\frac{13}{126}\right)\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{31}{126}\right)\) | \(e\left(\frac{53}{126}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{13}{42}\right)\) |
| \(\chi_{3724}(2483,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{126}\right)\) | \(e\left(\frac{95}{126}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{23}{126}\right)\) | \(e\left(\frac{19}{126}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{11}{42}\right)\) |
| \(\chi_{3724}(2651,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{126}\right)\) | \(e\left(\frac{17}{126}\right)\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{89}{126}\right)\) | \(e\left(\frac{79}{126}\right)\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{17}{42}\right)\) |
| \(\chi_{3724}(2663,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{126}\right)\) | \(e\left(\frac{103}{126}\right)\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{13}{126}\right)\) | \(e\left(\frac{71}{126}\right)\) | \(e\left(\frac{40}{63}\right)\) | \(e\left(\frac{19}{42}\right)\) |
| \(\chi_{3724}(2719,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{126}\right)\) | \(e\left(\frac{55}{126}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{73}{126}\right)\) | \(e\left(\frac{11}{126}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{13}{42}\right)\) |
| \(\chi_{3724}(2903,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{126}\right)\) | \(e\left(\frac{47}{126}\right)\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{83}{126}\right)\) | \(e\left(\frac{85}{126}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{5}{42}\right)\) |
| \(\chi_{3724}(3015,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{126}\right)\) | \(e\left(\frac{23}{126}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{113}{126}\right)\) | \(e\left(\frac{55}{126}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{23}{42}\right)\) |
| \(\chi_{3724}(3111,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{126}\right)\) | \(e\left(\frac{97}{126}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{115}{126}\right)\) | \(e\left(\frac{95}{126}\right)\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{13}{42}\right)\) |
| \(\chi_{3724}(3183,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{126}\right)\) | \(e\left(\frac{71}{126}\right)\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{41}{63}\right)\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{53}{126}\right)\) | \(e\left(\frac{115}{126}\right)\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{29}{42}\right)\) |