from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1421, base_ring=CyclotomicField(84))
M = H._module
chi = DirichletCharacter(H, M([22,15]))
chi.galois_orbit()
[g,chi] = znchar(Mod(61,1421))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1421\) | |
| Conductor: | \(1421\) | 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1421}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{11}{84}\right)\) |
| \(\chi_{1421}(73,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{37}{84}\right)\) |
| \(\chi_{1421}(222,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{59}{84}\right)\) |
| \(\chi_{1421}(234,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{19}{84}\right)\) |
| \(\chi_{1421}(271,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{17}{84}\right)\) |
| \(\chi_{1421}(292,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{47}{84}\right)\) |
| \(\chi_{1421}(334,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{65}{84}\right)\) |
| \(\chi_{1421}(409,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{67}{84}\right)\) |
| \(\chi_{1421}(598,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{13}{84}\right)\) |
| \(\chi_{1421}(710,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{79}{84}\right)\) |
| \(\chi_{1421}(794,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{55}{84}\right)\) |
| \(\chi_{1421}(936,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{29}{84}\right)\) |
| \(\chi_{1421}(971,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{23}{84}\right)\) |
| \(\chi_{1421}(976,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{31}{84}\right)\) |
| \(\chi_{1421}(1025,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{73}{84}\right)\) |
| \(\chi_{1421}(1041,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{53}{84}\right)\) |
| \(\chi_{1421}(1062,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{41}{84}\right)\) |
| \(\chi_{1421}(1081,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{1}{84}\right)\) |
| \(\chi_{1421}(1181,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{71}{84}\right)\) |
| \(\chi_{1421}(1258,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{83}{84}\right)\) |
| \(\chi_{1421}(1326,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{43}{84}\right)\) |
| \(\chi_{1421}(1361,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{61}{84}\right)\) |
| \(\chi_{1421}(1389,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{25}{84}\right)\) |
| \(\chi_{1421}(1419,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{5}{84}\right)\) |