from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5445, base_ring=CyclotomicField(4))
M = H._module
chi = DirichletCharacter(H, M([0,1,0]))
pari: [g,chi] = znchar(Mod(4357,5445))
Basic properties
Modulus: | \(5445\) | |
Conductor: | \(5\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(4\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{5}(2,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5445.j
\(\chi_{5445}(3268,\cdot)\) \(\chi_{5445}(4357,\cdot)\)
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: | \(\Q(\sqrt{-1}) \) |
Fixed field: | \(\Q(\zeta_{5})\) |
Values on generators
\((3026,4357,3511)\) → \((1,i,1)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
\( \chi_{ 5445 }(4357, a) \) | \(-1\) | \(1\) | \(i\) | \(-1\) | \(i\) | \(-i\) | \(-i\) | \(-1\) | \(1\) | \(i\) | \(-1\) | \(-i\) |
sage: chi.jacobi_sum(n)