from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1071, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([8,4,9]))
pari: [g,chi] = znchar(Mod(1024,1071))
Basic properties
| Modulus: | \(1071\) | |
| Conductor: | \(1071\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(12\) | 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 1071.cg
\(\chi_{1071}(4,\cdot)\) \(\chi_{1071}(268,\cdot)\) \(\chi_{1071}(319,\cdot)\) \(\chi_{1071}(1024,\cdot)\)
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: | \(\Q(\zeta_{12})\) |
| Fixed field: | 12.12.29428267022381449968353937.2 |
Values on generators
\((596,766,190)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{1}{3}\right),-i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(19\) | \(20\) |
| \( \chi_{ 1071 }(1024, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-i\) | \(-1\) | \(e\left(\frac{7}{12}\right)\) | \(i\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{12}\right)\) |
sage: chi.jacobi_sum(n)