from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(287, base_ring=CyclotomicField(8))
M = H._module
chi = DirichletCharacter(H, M([0,3]))
chi.galois_orbit()
[g,chi] = znchar(Mod(85,287))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(287\) | |
Conductor: | \(41\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(8\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from 41.e | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
Field of values: | \(\Q(\zeta_{8})\) |
Fixed field: | 8.0.194754273881.1 |
Characters in Galois orbit
Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{287}(85,\cdot)\) | \(-1\) | \(1\) | \(-i\) | \(e\left(\frac{5}{8}\right)\) | \(-1\) | \(i\) | \(e\left(\frac{3}{8}\right)\) | \(i\) | \(i\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) |
\(\chi_{287}(120,\cdot)\) | \(-1\) | \(1\) | \(-i\) | \(e\left(\frac{1}{8}\right)\) | \(-1\) | \(i\) | \(e\left(\frac{7}{8}\right)\) | \(i\) | \(i\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) |
\(\chi_{287}(232,\cdot)\) | \(-1\) | \(1\) | \(i\) | \(e\left(\frac{7}{8}\right)\) | \(-1\) | \(-i\) | \(e\left(\frac{1}{8}\right)\) | \(-i\) | \(-i\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) |
\(\chi_{287}(260,\cdot)\) | \(-1\) | \(1\) | \(i\) | \(e\left(\frac{3}{8}\right)\) | \(-1\) | \(-i\) | \(e\left(\frac{5}{8}\right)\) | \(-i\) | \(-i\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) |