from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6384, base_ring=CyclotomicField(6))
M = H._module
chi = DirichletCharacter(H, M([3,3,3,3,5]))
pari: [g,chi] = znchar(Mod(3527,6384))
Basic properties
Modulus: | \(6384\) | |
Conductor: | \(3192\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(6\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{3192}(1931,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6384.en
\(\chi_{6384}(3527,\cdot)\) \(\chi_{6384}(5879,\cdot)\)
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: | \(\Q(\sqrt{-3}) \) |
Fixed field: | 6.6.11740750253568.1 |
Values on generators
\((799,4789,2129,913,1009)\) → \((-1,-1,-1,-1,e\left(\frac{5}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 6384 }(3527, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(-1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) |
sage: chi.jacobi_sum(n)