from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(586971, base_ring=CyclotomicField(5082))
M = H._module
chi = DirichletCharacter(H, M([3388,1089,672]))
chi.galois_orbit()
[g,chi] = znchar(Mod(34,586971))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(586971\) | |
| Conductor: | \(586971\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(5082\) | 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: | $\Q(\zeta_{2541})$ |
| Fixed field: | Number field defined by a degree 5082 polynomial (not computed) |
First 3 of 1320 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{586971}(34,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{941}{2541}\right)\) | \(e\left(\frac{1882}{2541}\right)\) | \(e\left(\frac{4463}{5082}\right)\) | \(e\left(\frac{94}{847}\right)\) | \(e\left(\frac{421}{1694}\right)\) | \(e\left(\frac{1091}{5082}\right)\) | \(e\left(\frac{1223}{2541}\right)\) | \(e\left(\frac{647}{1694}\right)\) | \(e\left(\frac{93}{242}\right)\) | \(e\left(\frac{3145}{5082}\right)\) |
| \(\chi_{586971}(265,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{895}{2541}\right)\) | \(e\left(\frac{1790}{2541}\right)\) | \(e\left(\frac{3991}{5082}\right)\) | \(e\left(\frac{48}{847}\right)\) | \(e\left(\frac{233}{1694}\right)\) | \(e\left(\frac{2161}{5082}\right)\) | \(e\left(\frac{1039}{2541}\right)\) | \(e\left(\frac{853}{1694}\right)\) | \(e\left(\frac{117}{242}\right)\) | \(e\left(\frac{2489}{5082}\right)\) |
| \(\chi_{586971}(958,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{988}{2541}\right)\) | \(e\left(\frac{1976}{2541}\right)\) | \(e\left(\frac{1189}{5082}\right)\) | \(e\left(\frac{141}{847}\right)\) | \(e\left(\frac{1055}{1694}\right)\) | \(e\left(\frac{1213}{5082}\right)\) | \(e\left(\frac{1411}{2541}\right)\) | \(e\left(\frac{547}{1694}\right)\) | \(e\left(\frac{79}{242}\right)\) | \(e\left(\frac{59}{5082}\right)\) |