from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(185193, base_ring=CyclotomicField(6498))
M = H._module
chi = DirichletCharacter(H, M([361,3]))
pari: [g,chi] = znchar(Mod(144047,185193))
Basic properties
| Modulus: | \(185193\) | |
| Conductor: | \(185193\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(6498\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 185193.hs
\(\chi_{185193}(50,\cdot)\) \(\chi_{185193}(65,\cdot)\) \(\chi_{185193}(221,\cdot)\) \(\chi_{185193}(236,\cdot)\) \(\chi_{185193}(392,\cdot)\) \(\chi_{185193}(407,\cdot)\) \(\chi_{185193}(563,\cdot)\) \(\chi_{185193}(578,\cdot)\) \(\chi_{185193}(734,\cdot)\) \(\chi_{185193}(749,\cdot)\) \(\chi_{185193}(905,\cdot)\) \(\chi_{185193}(920,\cdot)\) \(\chi_{185193}(1076,\cdot)\) \(\chi_{185193}(1091,\cdot)\) \(\chi_{185193}(1247,\cdot)\) \(\chi_{185193}(1262,\cdot)\) \(\chi_{185193}(1418,\cdot)\) \(\chi_{185193}(1433,\cdot)\)
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
\((6860,178336)\) → \((e\left(\frac{1}{18}\right),e\left(\frac{1}{2166}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 185193 }(144047, a) \) | \(1\) | \(1\) | \(e\left(\frac{182}{3249}\right)\) | \(e\left(\frac{364}{3249}\right)\) | \(e\left(\frac{3869}{6498}\right)\) | \(e\left(\frac{377}{3249}\right)\) | \(e\left(\frac{182}{1083}\right)\) | \(e\left(\frac{1411}{2166}\right)\) | \(e\left(\frac{2947}{6498}\right)\) | \(e\left(\frac{6269}{6498}\right)\) | \(e\left(\frac{559}{3249}\right)\) | \(e\left(\frac{728}{3249}\right)\) |
sage: chi.jacobi_sum(n)