from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(39, base_ring=CyclotomicField(4))
M = H._module
chi = DirichletCharacter(H, M([2,1]))
pari: [g,chi] = znchar(Mod(8,39))
Basic properties
Modulus: | \(39\) | |
Conductor: | \(39\) | 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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 39.f
\(\chi_{39}(5,\cdot)\) \(\chi_{39}(8,\cdot)\)
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: | 4.4.19773.1 |
Values on generators
\((14,28)\) → \((-1,i)\)
Values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
\( \chi_{ 39 }(8, a) \) | \(1\) | \(1\) | \(-i\) | \(-1\) | \(-i\) | \(-i\) | \(i\) | \(-1\) | \(i\) | \(-1\) | \(1\) | \(1\) |
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)