sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(833, base_ring=CyclotomicField(336))
M = H._module
chi = DirichletCharacter(H, M([88,63]))
pari:[g,chi] = znchar(Mod(61,833))
Modulus: | \(833\) | |
Conductor: | \(833\) |
sage:chi.conductor()
pari:znconreyconductor(g,chi)
|
Order: | \(336\) |
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)
|
\(\chi_{833}(3,\cdot)\)
\(\chi_{833}(5,\cdot)\)
\(\chi_{833}(10,\cdot)\)
\(\chi_{833}(12,\cdot)\)
\(\chi_{833}(24,\cdot)\)
\(\chi_{833}(40,\cdot)\)
\(\chi_{833}(45,\cdot)\)
\(\chi_{833}(54,\cdot)\)
\(\chi_{833}(61,\cdot)\)
\(\chi_{833}(73,\cdot)\)
\(\chi_{833}(75,\cdot)\)
\(\chi_{833}(82,\cdot)\)
\(\chi_{833}(96,\cdot)\)
\(\chi_{833}(108,\cdot)\)
\(\chi_{833}(122,\cdot)\)
\(\chi_{833}(124,\cdot)\)
\(\chi_{833}(131,\cdot)\)
\(\chi_{833}(143,\cdot)\)
\(\chi_{833}(150,\cdot)\)
\(\chi_{833}(159,\cdot)\)
\(\chi_{833}(164,\cdot)\)
\(\chi_{833}(173,\cdot)\)
\(\chi_{833}(180,\cdot)\)
\(\chi_{833}(192,\cdot)\)
\(\chi_{833}(194,\cdot)\)
\(\chi_{833}(199,\cdot)\)
\(\chi_{833}(201,\cdot)\)
\(\chi_{833}(241,\cdot)\)
\(\chi_{833}(243,\cdot)\)
\(\chi_{833}(248,\cdot)\)
...
sage:chi.galois_orbit()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((52,785)\) → \((e\left(\frac{11}{42}\right),e\left(\frac{3}{16}\right))\)
\(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
\( \chi_{ 833 }(61, a) \) |
\(1\) | \(1\) | \(e\left(\frac{73}{168}\right)\) | \(e\left(\frac{151}{336}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{179}{336}\right)\) | \(e\left(\frac{99}{112}\right)\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{151}{168}\right)\) | \(e\left(\frac{325}{336}\right)\) | \(e\left(\frac{265}{336}\right)\) | \(e\left(\frac{107}{336}\right)\) |
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)