from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2040, base_ring=CyclotomicField(2))
M = H._module
chi = DirichletCharacter(H, M([0,1,1,1,1]))
pari: [g,chi] = znchar(Mod(509,2040))
Kronecker symbol representation
sage: kronecker_character(-2040)
pari: znchartokronecker(g,chi)
\(\displaystyle\left(\frac{-2040}{\bullet}\right)\)
Basic properties
Modulus: | \(2040\) | |
Conductor: | \(2040\) | 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 2040.v
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: | \(\Q(\sqrt{-510}) \) |
Values on generators
\((511,1021,1361,817,241)\) → \((1,-1,-1,-1,-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 2040 }(509, a) \) | \(-1\) | \(1\) | \(1\) | \(-1\) | \(1\) | \(-1\) | \(-1\) | \(-1\) | \(-1\) | \(-1\) | \(1\) | \(1\) |
sage: chi.jacobi_sum(n)