![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(403, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,1]))
![Copy content]()
pari:[g,chi] = znchar(Mod(313,403))
\(\chi_{403}(53,\cdot)\)
\(\chi_{403}(79,\cdot)\)
\(\chi_{403}(105,\cdot)\)
\(\chi_{403}(261,\cdot)\)
\(\chi_{403}(300,\cdot)\)
\(\chi_{403}(313,\cdot)\)
\(\chi_{403}(352,\cdot)\)
\(\chi_{403}(365,\cdot)\)
![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 ]
\((249,313)\) → \((1,e\left(\frac{1}{30}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 403 }(313, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{23}{30}\right)\) |
![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)
