from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1045, base_ring=CyclotomicField(4))
M = H._module
chi = DirichletCharacter(H, M([1,2,2]))
pari: [g,chi] = znchar(Mod(417,1045))
Basic properties
| Modulus: | \(1045\) | |
| Conductor: | \(1045\) | 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 1045.k
\(\chi_{1045}(208,\cdot)\) \(\chi_{1045}(417,\cdot)\)
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: | 4.0.5460125.4 |
Values on generators
\((837,761,496)\) → \((i,-1,-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 1045 }(417, a) \) | \(-1\) | \(1\) | \(i\) | \(i\) | \(-1\) | \(-1\) | \(-i\) | \(-i\) | \(-1\) | \(-i\) | \(-i\) | \(1\) |
sage: chi.jacobi_sum(n)