from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3724, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([0,87,112]))
chi.galois_orbit()
[g,chi] = znchar(Mod(5,3724))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3724\) | |
| Conductor: | \(931\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(126\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 931.cj | 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 | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(23\) | \(25\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3724}(5,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{126}\right)\) | \(e\left(\frac{31}{126}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{29}{126}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{19}{126}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{31}{42}\right)\) |
| \(\chi_{3724}(101,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{126}\right)\) | \(e\left(\frac{17}{126}\right)\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{85}{126}\right)\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{47}{126}\right)\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{17}{42}\right)\) |
| \(\chi_{3724}(213,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{126}\right)\) | \(e\left(\frac{5}{126}\right)\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{25}{126}\right)\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{125}{126}\right)\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{5}{42}\right)\) |
| \(\chi_{3724}(397,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{115}{126}\right)\) | \(e\left(\frac{115}{126}\right)\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{71}{126}\right)\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{103}{126}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{31}{42}\right)\) |
| \(\chi_{3724}(453,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{126}\right)\) | \(e\left(\frac{19}{126}\right)\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{95}{126}\right)\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{97}{126}\right)\) | \(e\left(\frac{25}{63}\right)\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{19}{42}\right)\) |
| \(\chi_{3724}(465,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{126}\right)\) | \(e\left(\frac{83}{126}\right)\) | \(e\left(\frac{20}{63}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{37}{126}\right)\) | \(e\left(\frac{20}{63}\right)\) | \(e\left(\frac{59}{126}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{20}{63}\right)\) | \(e\left(\frac{41}{42}\right)\) |
| \(\chi_{3724}(537,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{85}{126}\right)\) | \(e\left(\frac{85}{126}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{47}{126}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{109}{126}\right)\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{1}{42}\right)\) |
| \(\chi_{3724}(633,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{126}\right)\) | \(e\left(\frac{107}{126}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{31}{126}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{29}{126}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{23}{42}\right)\) |
| \(\chi_{3724}(745,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{95}{126}\right)\) | \(e\left(\frac{95}{126}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{97}{126}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{107}{126}\right)\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{11}{42}\right)\) |
| \(\chi_{3724}(929,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{126}\right)\) | \(e\left(\frac{43}{126}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{89}{126}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{67}{126}\right)\) | \(e\left(\frac{40}{63}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{1}{42}\right)\) |
| \(\chi_{3724}(985,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{126}\right)\) | \(e\left(\frac{73}{126}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{113}{126}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{61}{126}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{31}{42}\right)\) |
| \(\chi_{3724}(997,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{126}\right)\) | \(e\left(\frac{47}{126}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{109}{126}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{41}{126}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{5}{42}\right)\) |
| \(\chi_{3724}(1069,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{126}\right)\) | \(e\left(\frac{13}{126}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{65}{126}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{73}{126}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{13}{42}\right)\) |
| \(\chi_{3724}(1165,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{126}\right)\) | \(e\left(\frac{71}{126}\right)\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{103}{126}\right)\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{11}{126}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{29}{42}\right)\) |
| \(\chi_{3724}(1277,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{126}\right)\) | \(e\left(\frac{59}{126}\right)\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{43}{126}\right)\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{89}{126}\right)\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{17}{42}\right)\) |
| \(\chi_{3724}(1461,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{126}\right)\) | \(e\left(\frac{97}{126}\right)\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{107}{126}\right)\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{31}{126}\right)\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{13}{42}\right)\) |
| \(\chi_{3724}(1517,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{126}\right)\) | \(e\left(\frac{1}{126}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{5}{126}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{25}{126}\right)\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{1}{42}\right)\) |
| \(\chi_{3724}(1529,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{126}\right)\) | \(e\left(\frac{11}{126}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{55}{126}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{23}{126}\right)\) | \(e\left(\frac{41}{63}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{11}{42}\right)\) |
| \(\chi_{3724}(1601,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{126}\right)\) | \(e\left(\frac{67}{126}\right)\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{83}{126}\right)\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{37}{126}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{25}{42}\right)\) |
| \(\chi_{3724}(1809,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{126}\right)\) | \(e\left(\frac{23}{126}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{115}{126}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{71}{126}\right)\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{23}{42}\right)\) |
| \(\chi_{3724}(1993,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{126}\right)\) | \(e\left(\frac{25}{126}\right)\) | \(e\left(\frac{25}{63}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{125}{126}\right)\) | \(e\left(\frac{25}{63}\right)\) | \(e\left(\frac{121}{126}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{25}{63}\right)\) | \(e\left(\frac{25}{42}\right)\) |
| \(\chi_{3724}(2049,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{126}\right)\) | \(e\left(\frac{55}{126}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{23}{126}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{115}{126}\right)\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{13}{42}\right)\) |
| \(\chi_{3724}(2061,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{126}\right)\) | \(e\left(\frac{101}{126}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{1}{126}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{5}{126}\right)\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{17}{42}\right)\) |
| \(\chi_{3724}(2133,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{121}{126}\right)\) | \(e\left(\frac{121}{126}\right)\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{101}{126}\right)\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{1}{126}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{37}{42}\right)\) |
| \(\chi_{3724}(2229,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{125}{126}\right)\) | \(e\left(\frac{125}{126}\right)\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{121}{126}\right)\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{101}{126}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{41}{42}\right)\) |
| \(\chi_{3724}(2341,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{126}\right)\) | \(e\left(\frac{113}{126}\right)\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{61}{126}\right)\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{53}{126}\right)\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{29}{42}\right)\) |
| \(\chi_{3724}(2525,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{126}\right)\) | \(e\left(\frac{79}{126}\right)\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{17}{126}\right)\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{85}{126}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{37}{42}\right)\) |
| \(\chi_{3724}(2581,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{126}\right)\) | \(e\left(\frac{109}{126}\right)\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{41}{126}\right)\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{79}{126}\right)\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{25}{42}\right)\) |
| \(\chi_{3724}(2593,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{126}\right)\) | \(e\left(\frac{65}{126}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{73}{126}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{113}{126}\right)\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{23}{42}\right)\) |
| \(\chi_{3724}(2761,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{126}\right)\) | \(e\left(\frac{89}{126}\right)\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{67}{126}\right)\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{83}{126}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{5}{42}\right)\) |
| \(\chi_{3724}(3113,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{126}\right)\) | \(e\left(\frac{37}{126}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{59}{126}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{43}{126}\right)\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{37}{42}\right)\) |