from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6240, base_ring=CyclotomicField(8))
M = H._module
chi = DirichletCharacter(H, M([0,1,4,0,6]))
pari: [g,chi] = znchar(Mod(2501,6240))
Basic properties
| Modulus: | \(6240\) | |
| Conductor: | \(1248\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(8\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1248}(5,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6240.gw
\(\chi_{6240}(2501,\cdot)\) \(\chi_{6240}(2621,\cdot)\) \(\chi_{6240}(5621,\cdot)\) \(\chi_{6240}(5741,\cdot)\)
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: | 8.8.839604965361057792.2 |
Values on generators
\((1951,2341,2081,2497,5761)\) → \((1,e\left(\frac{1}{8}\right),-1,1,-i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 6240 }(2501, a) \) | \(1\) | \(1\) | \(-1\) | \(e\left(\frac{3}{8}\right)\) | \(-1\) | \(e\left(\frac{5}{8}\right)\) | \(-i\) | \(e\left(\frac{7}{8}\right)\) | \(-i\) | \(e\left(\frac{3}{8}\right)\) | \(1\) | \(e\left(\frac{1}{8}\right)\) |
sage: chi.jacobi_sum(n)