![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(58, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([23]))
![Copy content]()
pari:[g,chi] = znchar(Mod(39,58))
\(\chi_{58}(3,\cdot)\)
\(\chi_{58}(11,\cdot)\)
\(\chi_{58}(15,\cdot)\)
\(\chi_{58}(19,\cdot)\)
\(\chi_{58}(21,\cdot)\)
\(\chi_{58}(27,\cdot)\)
\(\chi_{58}(31,\cdot)\)
\(\chi_{58}(37,\cdot)\)
\(\chi_{58}(39,\cdot)\)
\(\chi_{58}(43,\cdot)\)
\(\chi_{58}(47,\cdot)\)
\(\chi_{58}(55,\cdot)\)
![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 ]
\(31\) → \(e\left(\frac{23}{28}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 58 }(39, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(i\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{27}{28}\right)\) |
![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)
