from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8820, base_ring=CyclotomicField(84))
M = H._module
chi = DirichletCharacter(H, M([0,0,21,64]))
chi.galois_orbit()
[g,chi] = znchar(Mod(37,8820))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8820\) | |
| Conductor: | \(245\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(84\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 245.w | 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: | $\Q(\zeta_{84})$ |
| Fixed field: | Number field defined by a degree 84 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8820}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{9}{28}\right)\) |
| \(\chi_{8820}(613,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{15}{28}\right)\) |
| \(\chi_{8820}(793,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{15}{28}\right)\) |
| \(\chi_{8820}(1117,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{17}{28}\right)\) |
| \(\chi_{8820}(1297,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{5}{28}\right)\) |
| \(\chi_{8820}(1873,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{27}{28}\right)\) |
| \(\chi_{8820}(2053,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{11}{28}\right)\) |
| \(\chi_{8820}(2377,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{1}{28}\right)\) |
| \(\chi_{8820}(2557,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{1}{28}\right)\) |
| \(\chi_{8820}(3133,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{11}{28}\right)\) |
| \(\chi_{8820}(3637,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{13}{28}\right)\) |
| \(\chi_{8820}(3817,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{25}{28}\right)\) |
| \(\chi_{8820}(4393,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{23}{28}\right)\) |
| \(\chi_{8820}(4573,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{3}{28}\right)\) |
| \(\chi_{8820}(4897,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{25}{28}\right)\) |
| \(\chi_{8820}(5833,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{27}{28}\right)\) |
| \(\chi_{8820}(6157,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{9}{28}\right)\) |
| \(\chi_{8820}(6337,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{17}{28}\right)\) |
| \(\chi_{8820}(6913,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{19}{28}\right)\) |
| \(\chi_{8820}(7093,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{23}{28}\right)\) |
| \(\chi_{8820}(7597,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{13}{28}\right)\) |
| \(\chi_{8820}(8173,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{3}{28}\right)\) |
| \(\chi_{8820}(8353,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{19}{28}\right)\) |
| \(\chi_{8820}(8677,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{5}{28}\right)\) |