sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(185, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([27,16]))
pari:[g,chi] = znchar(Mod(83,185))
Modulus: | \(185\) | |
Conductor: | \(185\) |
sage:chi.conductor()
pari:znconreyconductor(g,chi)
|
Order: | \(36\) |
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)
|
\(\chi_{185}(7,\cdot)\)
\(\chi_{185}(12,\cdot)\)
\(\chi_{185}(33,\cdot)\)
\(\chi_{185}(53,\cdot)\)
\(\chi_{185}(83,\cdot)\)
\(\chi_{185}(107,\cdot)\)
\(\chi_{185}(108,\cdot)\)
\(\chi_{185}(118,\cdot)\)
\(\chi_{185}(123,\cdot)\)
\(\chi_{185}(127,\cdot)\)
\(\chi_{185}(157,\cdot)\)
\(\chi_{185}(182,\cdot)\)
sage:chi.galois_orbit()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((112,76)\) → \((-i,e\left(\frac{4}{9}\right))\)
\(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
\( \chi_{ 185 }(83, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(1\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{5}{36}\right)\) |
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)