from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(100155, base_ring=CyclotomicField(2))
M = H._module
chi = DirichletCharacter(H, M([1,1,1,1]))
pari: [g,chi] = znchar(Mod(100154,100155))
Kronecker symbol representation
sage: kronecker_character(-100155)
pari: znchartokronecker(g,chi)
\(\displaystyle\left(\frac{-100155}{\bullet}\right)\)
Basic properties
Modulus: | \(100155\) | |
Conductor: | \(100155\) | 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 100155.h
\(\chi_{100155}(100154,\cdot)\)
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: | \(\Q\) |
Fixed field: | \(\Q(\sqrt{-100155}) \) |
Values on generators
\((33386,20032,63736,66166)\) → \((-1,-1,-1,-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
\( \chi_{ 100155 }(100154, a) \) | \(-1\) | \(1\) | \(-1\) | \(1\) | \(1\) | \(-1\) | \(1\) | \(-1\) | \(1\) | \(1\) | \(-1\) | \(-1\) |
sage: chi.jacobi_sum(n)