from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8820, base_ring=CyclotomicField(84))
M = H._module
chi = DirichletCharacter(H, M([42,56,63,12]))
chi.galois_orbit()
[g,chi] = znchar(Mod(43,8820))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8820\) | |
| Conductor: | \(8820\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(84\) | 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: | $\Q(\zeta_{84})$ |
| Fixed field: | Number field defined by a degree 84 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8820}(43,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(1\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{23}{84}\right)\) |
| \(\chi_{8820}(463,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(1\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{43}{84}\right)\) |
| \(\chi_{8820}(547,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(1\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{5}{84}\right)\) |
| \(\chi_{8820}(967,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(1\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{25}{84}\right)\) |
| \(\chi_{8820}(1303,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(1\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{83}{84}\right)\) |
| \(\chi_{8820}(1723,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(1\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{19}{84}\right)\) |
| \(\chi_{8820}(1807,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(1\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{65}{84}\right)\) |
| \(\chi_{8820}(2227,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(1\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{1}{84}\right)\) |
| \(\chi_{8820}(2563,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(1\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{59}{84}\right)\) |
| \(\chi_{8820}(2983,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(1\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{79}{84}\right)\) |
| \(\chi_{8820}(3067,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(1\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{41}{84}\right)\) |
| \(\chi_{8820}(3487,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(1\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{61}{84}\right)\) |
| \(\chi_{8820}(4243,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(1\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{55}{84}\right)\) |
| \(\chi_{8820}(4327,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(1\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{17}{84}\right)\) |
| \(\chi_{8820}(4747,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(1\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{37}{84}\right)\) |
| \(\chi_{8820}(5083,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(1\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{11}{84}\right)\) |
| \(\chi_{8820}(5503,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(1\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{31}{84}\right)\) |
| \(\chi_{8820}(6007,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(1\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{13}{84}\right)\) |
| \(\chi_{8820}(6343,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(1\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{71}{84}\right)\) |
| \(\chi_{8820}(6847,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(1\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{53}{84}\right)\) |
| \(\chi_{8820}(7267,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(1\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{73}{84}\right)\) |
| \(\chi_{8820}(7603,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(1\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{47}{84}\right)\) |
| \(\chi_{8820}(8023,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(1\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{67}{84}\right)\) |
| \(\chi_{8820}(8107,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(1\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{29}{84}\right)\) |