from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8640, base_ring=CyclotomicField(8))
M = H._module
chi = DirichletCharacter(H, M([0,7,4,2]))
pari: [g,chi] = znchar(Mod(2537,8640))
Basic properties
| Modulus: | \(8640\) | |
| Conductor: | \(480\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(8\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{480}(77,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8640.cr
\(\chi_{8640}(2537,\cdot)\) \(\chi_{8640}(3833,\cdot)\) \(\chi_{8640}(6857,\cdot)\) \(\chi_{8640}(8153,\cdot)\)
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: | 8.8.2717908992000000.2 |
Values on generators
\((2431,3781,6401,3457)\) → \((1,e\left(\frac{7}{8}\right),-1,i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 8640 }(2537, a) \) | \(1\) | \(1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(i\) | \(e\left(\frac{5}{8}\right)\) | \(-1\) | \(e\left(\frac{5}{8}\right)\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)