sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(90, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([8,9]))
pari:[g,chi] = znchar(Mod(43,90))
\(\chi_{90}(7,\cdot)\)
\(\chi_{90}(13,\cdot)\)
\(\chi_{90}(43,\cdot)\)
\(\chi_{90}(67,\cdot)\)
sage:chi.galois_orbit()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((11,37)\) → \((e\left(\frac{2}{3}\right),-i)\)
\(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 90 }(43, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(-i\) | \(-1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-i\) | \(e\left(\frac{1}{3}\right)\) |
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)