![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(756, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([0,7,6]))
![Copy content]()
pari:[g,chi] = znchar(Mod(317,756))
\(\chi_{756}(65,\cdot)\)
\(\chi_{756}(221,\cdot)\)
\(\chi_{756}(317,\cdot)\)
\(\chi_{756}(473,\cdot)\)
\(\chi_{756}(569,\cdot)\)
\(\chi_{756}(725,\cdot)\)
![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 ]
\((379,29,325)\) → \((1,e\left(\frac{7}{18}\right),e\left(\frac{1}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 756 }(317, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(1\) |
![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)
