from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3120, base_ring=CyclotomicField(4))
M = H._module
chi = DirichletCharacter(H, M([0,0,0,1,1]))
pari: [g,chi] = znchar(Mod(2257,3120))
Basic properties
Modulus: | \(3120\) | |
Conductor: | \(65\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(4\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{65}(47,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3120.fc
\(\chi_{3120}(2257,\cdot)\) \(\chi_{3120}(3073,\cdot)\)
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: | 4.4.274625.2 |
Values on generators
\((1951,2341,2081,2497,2641)\) → \((1,1,1,i,i)\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 3120 }(2257, a) \) | \(1\) | \(1\) | \(1\) | \(-i\) | \(-i\) | \(-i\) | \(i\) | \(-1\) | \(i\) | \(1\) | \(i\) | \(i\) |
sage: chi.jacobi_sum(n)