from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(352, base_ring=CyclotomicField(4))
M = H._module
chi = DirichletCharacter(H, M([0,1,0]))
chi.galois_orbit()
[g,chi] = znchar(Mod(89,352))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(352\) | |
Conductor: | \(16\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(4\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from 16.e | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
Field of values: | \(\mathbb{Q}(i)\) |
Fixed field: | \(\Q(\zeta_{16})^+\) |
Characters in Galois orbit
Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{352}(89,\cdot)\) | \(1\) | \(1\) | \(-i\) | \(i\) | \(-1\) | \(-1\) | \(-i\) | \(1\) | \(1\) | \(-i\) | \(i\) | \(-1\) |
\(\chi_{352}(265,\cdot)\) | \(1\) | \(1\) | \(i\) | \(-i\) | \(-1\) | \(-1\) | \(i\) | \(1\) | \(1\) | \(i\) | \(-i\) | \(-1\) |