from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(45, base_ring=CyclotomicField(4))
M = H._module
chi = DirichletCharacter(H, M([2,3]))
pari: [g,chi] = znchar(Mod(8,45))
Basic properties
Modulus: | \(45\) | |
Conductor: | \(15\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(4\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{15}(8,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 45.f
\(\chi_{45}(8,\cdot)\) \(\chi_{45}(17,\cdot)\)
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: | \(\Q(\zeta_{15})^+\) |
Values on generators
\((11,37)\) → \((-1,-i)\)
Values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 45 }(8, a) \) | \(1\) | \(1\) | \(i\) | \(-1\) | \(-i\) | \(-i\) | \(-1\) | \(i\) | \(1\) | \(1\) | \(i\) | \(-1\) |
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)