from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4508, base_ring=CyclotomicField(6))
M = H._module
chi = DirichletCharacter(H, M([3,1,0]))
pari: [g,chi] = znchar(Mod(1795,4508))
Basic properties
Modulus: | \(4508\) | |
Conductor: | \(28\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(6\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{28}(3,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4508.k
\(\chi_{4508}(1795,\cdot)\) \(\chi_{4508}(1979,\cdot)\)
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: | \(\mathbb{Q}(\zeta_3)\) |
Fixed field: | \(\Q(\zeta_{28})^+\) |
Values on generators
\((2255,1473,1569)\) → \((-1,e\left(\frac{1}{6}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(25\) | \(27\) |
\( \chi_{ 4508 }(1795, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(-1\) | \(-1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)