![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(558, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([25,27]))
![Copy content]()
pari:[g,chi] = znchar(Mod(23,558))
\(\chi_{558}(23,\cdot)\)
\(\chi_{558}(29,\cdot)\)
\(\chi_{558}(77,\cdot)\)
\(\chi_{558}(209,\cdot)\)
\(\chi_{558}(263,\cdot)\)
\(\chi_{558}(275,\cdot)\)
\(\chi_{558}(401,\cdot)\)
\(\chi_{558}(461,\cdot)\)
![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 ]
\((497,127)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{9}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(35\) |
| \( \chi_{ 558 }(23, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{7}{10}\right)\) |
![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)
