from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3712, base_ring=CyclotomicField(8))
M = H._module
chi = DirichletCharacter(H, M([0,5,0]))
pari: [g,chi] = znchar(Mod(3249,3712))
Basic properties
Modulus: | \(3712\) | |
Conductor: | \(32\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(8\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{32}(21,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3712.v
\(\chi_{3712}(465,\cdot)\) \(\chi_{3712}(1393,\cdot)\) \(\chi_{3712}(2321,\cdot)\) \(\chi_{3712}(3249,\cdot)\)
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: | \(\Q(\zeta_{8})\) |
Fixed field: | \(\Q(\zeta_{32})^+\) |
Values on generators
\((639,2437,2177)\) → \((1,e\left(\frac{5}{8}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 3712 }(3249, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(i\) | \(-i\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(-1\) | \(-1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) |
sage: chi.jacobi_sum(n)