sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(547, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([2]))
pari:[g,chi] = znchar(Mod(521,547))
Modulus: | \(547\) | |
Conductor: | \(547\) |
sage:chi.conductor()
pari:znconreyconductor(g,chi)
|
Order: | \(39\) |
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)
|
\(\chi_{547}(11,\cdot)\)
\(\chi_{547}(21,\cdot)\)
\(\chi_{547}(47,\cdot)\)
\(\chi_{547}(54,\cdot)\)
\(\chi_{547}(96,\cdot)\)
\(\chi_{547}(121,\cdot)\)
\(\chi_{547}(129,\cdot)\)
\(\chi_{547}(136,\cdot)\)
\(\chi_{547}(181,\cdot)\)
\(\chi_{547}(199,\cdot)\)
\(\chi_{547}(217,\cdot)\)
\(\chi_{547}(231,\cdot)\)
\(\chi_{547}(233,\cdot)\)
\(\chi_{547}(239,\cdot)\)
\(\chi_{547}(296,\cdot)\)
\(\chi_{547}(302,\cdot)\)
\(\chi_{547}(325,\cdot)\)
\(\chi_{547}(402,\cdot)\)
\(\chi_{547}(419,\cdot)\)
\(\chi_{547}(441,\cdot)\)
\(\chi_{547}(445,\cdot)\)
\(\chi_{547}(464,\cdot)\)
\(\chi_{547}(488,\cdot)\)
\(\chi_{547}(521,\cdot)\)
sage:chi.galois_orbit()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\(2\) → \(e\left(\frac{1}{39}\right)\)
\(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 547 }(521, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{39}\right)\) | \(1\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(1\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{31}{39}\right)\) |
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)