from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3648, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([0,21,12,8]))
chi.galois_orbit()
[g,chi] = znchar(Mod(425,3648))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3648\) | |
| Conductor: | \(1824\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 1824.cw | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | \(\Q(\zeta_{24})\) |
| Fixed field: | Number field defined by a degree 24 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3648}(425,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{24}\right)\) | \(-i\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(1\) | \(e\left(\frac{11}{24}\right)\) |
| \(\chi_{3648}(809,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{24}\right)\) | \(-i\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(1\) | \(e\left(\frac{19}{24}\right)\) |
| \(\chi_{3648}(1337,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{24}\right)\) | \(i\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(1\) | \(e\left(\frac{17}{24}\right)\) |
| \(\chi_{3648}(1721,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{24}\right)\) | \(i\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(1\) | \(e\left(\frac{1}{24}\right)\) |
| \(\chi_{3648}(2249,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{24}\right)\) | \(-i\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(1\) | \(e\left(\frac{23}{24}\right)\) |
| \(\chi_{3648}(2633,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{24}\right)\) | \(-i\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(1\) | \(e\left(\frac{7}{24}\right)\) |
| \(\chi_{3648}(3161,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{24}\right)\) | \(i\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(1\) | \(e\left(\frac{5}{24}\right)\) |
| \(\chi_{3648}(3545,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{24}\right)\) | \(i\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(1\) | \(e\left(\frac{13}{24}\right)\) |