from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4029, base_ring=CyclotomicField(8))
M = H._module
chi = DirichletCharacter(H, M([4,1,4]))
pari: [g,chi] = znchar(Mod(1658,4029))
Basic properties
| Modulus: | \(4029\) | |
| Conductor: | \(4029\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(8\) | 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 4029.x
\(\chi_{4029}(236,\cdot)\) \(\chi_{4029}(1658,\cdot)\) \(\chi_{4029}(1895,\cdot)\) \(\chi_{4029}(3317,\cdot)\)
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: | Number field defined by a degree 8 polynomial |
Values on generators
\((2687,3556,3163)\) → \((-1,e\left(\frac{1}{8}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 4029 }(1658, a) \) | \(1\) | \(1\) | \(i\) | \(-1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(-i\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(-1\) | \(e\left(\frac{1}{8}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)