from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(441, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([14,24]))
chi.galois_orbit()
[g,chi] = znchar(Mod(22,441))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(441\) | |
| Conductor: | \(441\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(21\) | 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: | \(\Q(\zeta_{21})\) |
| Fixed field: | Number field defined by a degree 21 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{441}(22,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(1\) |
| \(\chi_{441}(43,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(1\) |
| \(\chi_{441}(85,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(1\) |
| \(\chi_{441}(106,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(1\) |
| \(\chi_{441}(169,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(1\) |
| \(\chi_{441}(211,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(1\) |
| \(\chi_{441}(232,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(1\) |
| \(\chi_{441}(274,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(1\) |
| \(\chi_{441}(337,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(1\) |
| \(\chi_{441}(358,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(1\) |
| \(\chi_{441}(400,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(1\) |
| \(\chi_{441}(421,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(1\) |