from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(555579, base_ring=CyclotomicField(6498))
M = H._module
chi = DirichletCharacter(H, M([5054,5859]))
pari: [g,chi] = znchar(Mod(37,555579))
Basic properties
| Modulus: | \(555579\) | |
| Conductor: | \(185193\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(6498\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{185193}(82345,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 555579.ji
\(\chi_{555579}(37,\cdot)\) \(\chi_{555579}(208,\cdot)\) \(\chi_{555579}(550,\cdot)\) \(\chi_{555579}(1063,\cdot)\) \(\chi_{555579}(1234,\cdot)\)
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: | $\Q(\zeta_{3249})$ |
| Fixed field: | Number field defined by a degree 6498 polynomial (not computed) |
Values on generators
\((192053,363529)\) → \((e\left(\frac{7}{9}\right),e\left(\frac{651}{722}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 555579 }(37, a) \) | \(-1\) | \(1\) | \(e\left(\frac{4415}{6498}\right)\) | \(e\left(\frac{1166}{3249}\right)\) | \(e\left(\frac{755}{3249}\right)\) | \(e\left(\frac{202}{3249}\right)\) | \(e\left(\frac{83}{2166}\right)\) | \(e\left(\frac{1975}{2166}\right)\) | \(e\left(\frac{1117}{3249}\right)\) | \(e\left(\frac{2569}{6498}\right)\) | \(e\left(\frac{4819}{6498}\right)\) | \(e\left(\frac{2332}{3249}\right)\) |
sage: chi.jacobi_sum(n)