from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(555579, base_ring=CyclotomicField(6498))
M = H._module
chi = DirichletCharacter(H, M([361,3]))
pari: [g,chi] = znchar(Mod(8,555579))
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 \(\chi_{185193}(144047,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 555579.km
\(\chi_{555579}(8,\cdot)\) \(\chi_{555579}(449,\cdot)\) \(\chi_{555579}(521,\cdot)\) \(\chi_{555579}(962,\cdot)\) \(\chi_{555579}(1034,\cdot)\) \(\chi_{555579}(1475,\cdot)\) \(\chi_{555579}(1547,\cdot)\)
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
\((192053,363529)\) → \((e\left(\frac{1}{18}\right),e\left(\frac{1}{2166}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 555579 }(8, a) \) | \(1\) | \(1\) | \(e\left(\frac{182}{3249}\right)\) | \(e\left(\frac{364}{3249}\right)\) | \(e\left(\frac{3869}{6498}\right)\) | \(e\left(\frac{377}{3249}\right)\) | \(e\left(\frac{182}{1083}\right)\) | \(e\left(\frac{1411}{2166}\right)\) | \(e\left(\frac{2947}{6498}\right)\) | \(e\left(\frac{6269}{6498}\right)\) | \(e\left(\frac{559}{3249}\right)\) | \(e\left(\frac{728}{3249}\right)\) |
sage: chi.jacobi_sum(n)