from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4080, base_ring=CyclotomicField(8))
M = H._module
chi = DirichletCharacter(H, M([0,0,4,2,7]))
pari: [g,chi] = znchar(Mod(257,4080))
Basic properties
| Modulus: | \(4080\) | |
| Conductor: | \(255\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(8\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{255}(2,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4080.gm
\(\chi_{4080}(257,\cdot)\) \(\chi_{4080}(1793,\cdot)\) \(\chi_{4080}(3857,\cdot)\) \(\chi_{4080}(3953,\cdot)\)
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: | 8.8.519334883015625.2 |
Values on generators
\((511,3061,1361,817,241)\) → \((1,1,-1,i,e\left(\frac{7}{8}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 4080 }(257, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(i\) | \(-i\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)