from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1312, base_ring=CyclotomicField(8))
M = H._module
chi = DirichletCharacter(H, M([4,6,5]))
pari: [g,chi] = znchar(Mod(55,1312))
Basic properties
| Modulus: | \(1312\) | |
| Conductor: | \(656\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(8\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{656}(547,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1312.x
\(\chi_{1312}(55,\cdot)\) \(\chi_{1312}(167,\cdot)\) \(\chi_{1312}(1063,\cdot)\) \(\chi_{1312}(1175,\cdot)\)
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: | \(\Q(\zeta_{8})\) |
| Fixed field: | 8.8.816858629956173824.3 |
Values on generators
\((575,165,129)\) → \((-1,-i,e\left(\frac{5}{8}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 1312 }(55, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(-1\) | \(e\left(\frac{3}{8}\right)\) | \(i\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)