sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1680, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([6,9,6,3,10]))
pari:[g,chi] = znchar(Mod(467,1680))
Modulus: | \(1680\) | |
Conductor: | \(1680\) |
sage:chi.conductor()
pari:znconreyconductor(g,chi)
|
Order: | \(12\) |
sage:chi.multiplicative_order()
pari:charorder(g,chi)
|
Real: | no |
Primitive: | yes |
sage:chi.is_primitive()
pari:#znconreyconductor(g,chi)==1
|
Minimal: | yes |
Parity: | even |
sage:chi.is_odd()
pari:zncharisodd(g,chi)
|
\(\chi_{1680}(227,\cdot)\)
\(\chi_{1680}(467,\cdot)\)
\(\chi_{1680}(1403,\cdot)\)
\(\chi_{1680}(1643,\cdot)\)
sage:chi.galois_orbit()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((1471,421,1121,337,241)\) → \((-1,-i,-1,i,e\left(\frac{5}{6}\right))\)
\(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 1680 }(467, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(-1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(i\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(1\) |
sage:chi.jacobi_sum(n)