from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(248, base_ring=CyclotomicField(2))
M = H._module
chi = DirichletCharacter(H, M([0,1,1]))
pari: [g,chi] = znchar(Mod(61,248))
Kronecker symbol representation
sage: kronecker_character(-248)
pari: znchartokronecker(g,chi)
\(\displaystyle\left(\frac{-248}{\bullet}\right)\)
Basic properties
| Modulus: | \(248\) | |
| Conductor: | \(248\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(2\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | yes | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 248.g
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: | \(\Q\) |
| Fixed field: | \(\Q(\sqrt{-62}) \) |
Values on generators
\((63,125,65)\) → \((1,-1,-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 248 }(61, a) \) | \(-1\) | \(1\) | \(1\) | \(-1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(-1\) | \(-1\) | \(-1\) | \(1\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)