from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(586971, base_ring=CyclotomicField(5082))
M = H._module
chi = DirichletCharacter(H, M([4235,4598,1218]))
pari: [g,chi] = znchar(Mod(23,586971))
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)
|
Galois orbit 586971.qh
\(\chi_{586971}(23,\cdot)\) \(\chi_{586971}(452,\cdot)\) \(\chi_{586971}(1409,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
Field of values: | $\Q(\zeta_{2541})$ |
Fixed field: | Number field defined by a degree 5082 polynomial (not computed) |
Values on generators
\((130439,179686,73207)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{19}{21}\right),e\left(\frac{29}{121}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
\( \chi_{ 586971 }(23, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1011}{1694}\right)\) | \(e\left(\frac{164}{847}\right)\) | \(e\left(\frac{2561}{5082}\right)\) | \(e\left(\frac{1339}{1694}\right)\) | \(e\left(\frac{256}{2541}\right)\) | \(e\left(\frac{932}{2541}\right)\) | \(e\left(\frac{328}{847}\right)\) | \(e\left(\frac{1151}{5082}\right)\) | \(e\left(\frac{302}{363}\right)\) | \(e\left(\frac{3545}{5082}\right)\) |
sage: chi.jacobi_sum(n)