from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5, base_ring=CyclotomicField(4))
M = H._module
chi = DirichletCharacter(H, M([1]))
pari: [g,chi] = znchar(Mod(2,5))
Basic properties
| Modulus: | \(5\) | |
| Conductor: | \(5\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(4\) | 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 5.c
\(\chi_{5}(2,\cdot)\) \(\chi_{5}(3,\cdot)\)
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: | \(\mathbb{Q}(i)\) |
| Fixed field: | \(\Q(\zeta_{5})\) |
Values on generators
\(2\) → \(i\)
Values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) |
| \( \chi_{ 5 }(2, a) \) | \(-1\) | \(1\) | \(i\) | \(-i\) |
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)
Additional information
This is the first example of a Dirichlet character whose values do not all lie in the field of rational numbers.
This also makes it the first example of a Dirichlet character whose Galois orbit is nontrivial.