![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(340, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,0,5]))
![Copy content]()
pari:[g,chi] = znchar(Mod(311,340))
\(\chi_{340}(11,\cdot)\)
\(\chi_{340}(31,\cdot)\)
\(\chi_{340}(71,\cdot)\)
\(\chi_{340}(91,\cdot)\)
\(\chi_{340}(131,\cdot)\)
\(\chi_{340}(211,\cdot)\)
\(\chi_{340}(231,\cdot)\)
\(\chi_{340}(311,\cdot)\)
![Copy content]()
sage:chi.galois_orbit()
![Copy content]()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((171,137,241)\) → \((-1,1,e\left(\frac{5}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 340 }(311, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(i\) | \(e\left(\frac{7}{8}\right)\) | \(-i\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) |
![Copy content]()
sage:chi.jacobi_sum(n)
![Copy content]()
sage:chi.gauss_sum(a)
![Copy content]()
pari:znchargauss(g,chi,a)
![Copy content]()
sage:chi.jacobi_sum(n)
![Copy content]()
sage:chi.kloosterman_sum(a,b)
