sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(539, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([40,42]))
pari:[g,chi] = znchar(Mod(191,539))
Modulus: | \(539\) | |
Conductor: | \(539\) |
sage:chi.conductor()
pari:znconreyconductor(g,chi)
|
Order: | \(105\) |
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)
|
\(\chi_{539}(4,\cdot)\)
\(\chi_{539}(9,\cdot)\)
\(\chi_{539}(16,\cdot)\)
\(\chi_{539}(25,\cdot)\)
\(\chi_{539}(37,\cdot)\)
\(\chi_{539}(53,\cdot)\)
\(\chi_{539}(58,\cdot)\)
\(\chi_{539}(60,\cdot)\)
\(\chi_{539}(81,\cdot)\)
\(\chi_{539}(86,\cdot)\)
\(\chi_{539}(93,\cdot)\)
\(\chi_{539}(102,\cdot)\)
\(\chi_{539}(114,\cdot)\)
\(\chi_{539}(130,\cdot)\)
\(\chi_{539}(135,\cdot)\)
\(\chi_{539}(137,\cdot)\)
\(\chi_{539}(158,\cdot)\)
\(\chi_{539}(163,\cdot)\)
\(\chi_{539}(170,\cdot)\)
\(\chi_{539}(179,\cdot)\)
\(\chi_{539}(191,\cdot)\)
\(\chi_{539}(207,\cdot)\)
\(\chi_{539}(212,\cdot)\)
\(\chi_{539}(235,\cdot)\)
\(\chi_{539}(240,\cdot)\)
\(\chi_{539}(247,\cdot)\)
\(\chi_{539}(256,\cdot)\)
\(\chi_{539}(268,\cdot)\)
\(\chi_{539}(284,\cdot)\)
\(\chi_{539}(289,\cdot)\)
...
sage:chi.galois_orbit()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((199,442)\) → \((e\left(\frac{4}{21}\right),e\left(\frac{1}{5}\right))\)
\(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
\( \chi_{ 539 }(191, a) \) |
\(1\) | \(1\) | \(e\left(\frac{16}{105}\right)\) | \(e\left(\frac{83}{105}\right)\) | \(e\left(\frac{32}{105}\right)\) | \(e\left(\frac{34}{105}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{61}{105}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{17}{35}\right)\) |
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)