from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(419, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([1]))
pari: [g,chi] = znchar(Mod(372,419))
Basic properties
Modulus: | \(419\) | |
Conductor: | \(419\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(38\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 419.f
\(\chi_{419}(40,\cdot)\) \(\chi_{419}(76,\cdot)\) \(\chi_{419}(89,\cdot)\) \(\chi_{419}(90,\cdot)\) \(\chi_{419}(113,\cdot)\) \(\chi_{419}(171,\cdot)\) \(\chi_{419}(204,\cdot)\) \(\chi_{419}(211,\cdot)\) \(\chi_{419}(220,\cdot)\) \(\chi_{419}(280,\cdot)\) \(\chi_{419}(283,\cdot)\) \(\chi_{419}(284,\cdot)\) \(\chi_{419}(305,\cdot)\) \(\chi_{419}(312,\cdot)\) \(\chi_{419}(359,\cdot)\) \(\chi_{419}(370,\cdot)\) \(\chi_{419}(372,\cdot)\) \(\chi_{419}(412,\cdot)\)
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: | \(\Q(\zeta_{19})\) |
Fixed field: | Number field defined by a degree 38 polynomial |
Values on generators
\(2\) → \(e\left(\frac{1}{38}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 419 }(372, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{21}{38}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)