sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(153, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,1]))
pari:[g,chi] = znchar(Mod(71,153))
\(\chi_{153}(44,\cdot)\)
\(\chi_{153}(62,\cdot)\)
\(\chi_{153}(71,\cdot)\)
\(\chi_{153}(80,\cdot)\)
\(\chi_{153}(107,\cdot)\)
\(\chi_{153}(116,\cdot)\)
\(\chi_{153}(125,\cdot)\)
\(\chi_{153}(143,\cdot)\)
sage:chi.galois_orbit()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((137,37)\) → \((-1,e\left(\frac{1}{16}\right))\)
\(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 153 }(71, a) \) |
\(1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(-i\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(i\) | \(e\left(\frac{1}{16}\right)\) | \(-1\) |
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)