sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(224, base_ring=CyclotomicField(8))
M = H._module
chi = DirichletCharacter(H, M([0,7,4]))
pari:[g,chi] = znchar(Mod(13,224))
Modulus: | \(224\) | |
Conductor: | \(224\) |
sage:chi.conductor()
pari:znconreyconductor(g,chi)
|
Order: | \(8\) |
sage:chi.multiplicative_order()
pari:charorder(g,chi)
|
Real: | no |
Primitive: | yes |
sage:chi.is_primitive()
pari:#znconreyconductor(g,chi)==1
|
Minimal: | yes |
Parity: | odd |
sage:chi.is_odd()
pari:zncharisodd(g,chi)
|
\(\chi_{224}(13,\cdot)\)
\(\chi_{224}(69,\cdot)\)
\(\chi_{224}(125,\cdot)\)
\(\chi_{224}(181,\cdot)\)
sage:chi.galois_orbit()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((127,197,129)\) → \((1,e\left(\frac{7}{8}\right),-1)\)
\(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
\( \chi_{ 224 }(13, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(i\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(i\) | \(-i\) |
sage:chi.jacobi_sum(n)
sage:chi.gauss_sum(a)
pari:znchargauss(g,chi,a)
sage:chi.jacobi_sum(n)
sage:chi.kloosterman_sum(a,b)