from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4800, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,5,0,2]))
chi.galois_orbit()
[g,chi] = znchar(Mod(79,4800))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4800\) | |
| Conductor: | \(400\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 400.bf | 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: | \(\Q(\zeta_{20})\) |
| Fixed field: | 20.0.20971520000000000000000000000000000000000.1 |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4800}(79,\cdot)\) | \(-1\) | \(1\) | \(-1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) |
| \(\chi_{4800}(559,\cdot)\) | \(-1\) | \(1\) | \(-1\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) |
| \(\chi_{4800}(1039,\cdot)\) | \(-1\) | \(1\) | \(-1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) |
| \(\chi_{4800}(1519,\cdot)\) | \(-1\) | \(1\) | \(-1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) |
| \(\chi_{4800}(2479,\cdot)\) | \(-1\) | \(1\) | \(-1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) |
| \(\chi_{4800}(2959,\cdot)\) | \(-1\) | \(1\) | \(-1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) |
| \(\chi_{4800}(3439,\cdot)\) | \(-1\) | \(1\) | \(-1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) |
| \(\chi_{4800}(3919,\cdot)\) | \(-1\) | \(1\) | \(-1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) |