![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(261, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,13]))
![Copy content]()
pari:[g,chi] = znchar(Mod(188,261))
\(\chi_{261}(8,\cdot)\)
\(\chi_{261}(26,\cdot)\)
\(\chi_{261}(44,\cdot)\)
\(\chi_{261}(89,\cdot)\)
\(\chi_{261}(98,\cdot)\)
\(\chi_{261}(134,\cdot)\)
\(\chi_{261}(143,\cdot)\)
\(\chi_{261}(188,\cdot)\)
\(\chi_{261}(206,\cdot)\)
\(\chi_{261}(224,\cdot)\)
\(\chi_{261}(242,\cdot)\)
\(\chi_{261}(251,\cdot)\)
![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 ]
\((146,118)\) → \((-1,e\left(\frac{13}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 261 }(188, a) \) |
\(1\) | \(1\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{6}{7}\right)\) |
![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)
