from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1764, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,7,9]))
chi.galois_orbit()
[g,chi] = znchar(Mod(83,1764))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1764\) | |
| Conductor: | \(1764\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | 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: | \(\Q(\zeta_{21})\) |
| Fixed field: | 42.0.12140927516266774666314598865493926760663766983493911125885658849232780316031818617435329218169600416415744.1 |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1764}(83,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(1\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{6}{7}\right)\) |
| \(\chi_{1764}(167,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(1\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{7}\right)\) |
| \(\chi_{1764}(335,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(1\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{3}{7}\right)\) |
| \(\chi_{1764}(419,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(1\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{7}\right)\) |
| \(\chi_{1764}(671,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(1\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{6}{7}\right)\) |
| \(\chi_{1764}(839,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(1\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{4}{7}\right)\) |
| \(\chi_{1764}(923,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(1\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{3}{7}\right)\) |
| \(\chi_{1764}(1091,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(1\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{7}\right)\) |
| \(\chi_{1764}(1343,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(1\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{7}\right)\) |
| \(\chi_{1764}(1427,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(1\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{4}{7}\right)\) |
| \(\chi_{1764}(1595,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(1\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{7}\right)\) |
| \(\chi_{1764}(1679,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(1\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{7}\right)\) |