![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(871, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,18]))
![Copy content]()
pari:[g,chi] = znchar(Mod(417,871))
\(\chi_{871}(14,\cdot)\)
\(\chi_{871}(40,\cdot)\)
\(\chi_{871}(92,\cdot)\)
\(\chi_{871}(131,\cdot)\)
\(\chi_{871}(196,\cdot)\)
\(\chi_{871}(417,\cdot)\)
\(\chi_{871}(560,\cdot)\)
\(\chi_{871}(612,\cdot)\)
\(\chi_{871}(625,\cdot)\)
\(\chi_{871}(729,\cdot)\)
![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 ]
\((470,404)\) → \((1,e\left(\frac{9}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 871 }(417, a) \) |
\(1\) | \(1\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) |
![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)
