![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(608, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,9,10]))
![Copy content]()
pari:[g,chi] = znchar(Mod(583,608))
\(\chi_{608}(71,\cdot)\)
\(\chi_{608}(135,\cdot)\)
\(\chi_{608}(167,\cdot)\)
\(\chi_{608}(231,\cdot)\)
\(\chi_{608}(279,\cdot)\)
\(\chi_{608}(295,\cdot)\)
\(\chi_{608}(375,\cdot)\)
\(\chi_{608}(439,\cdot)\)
\(\chi_{608}(471,\cdot)\)
\(\chi_{608}(535,\cdot)\)
\(\chi_{608}(583,\cdot)\)
\(\chi_{608}(599,\cdot)\)
![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 ]
\((191,229,97)\) → \((-1,i,e\left(\frac{5}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
| \( \chi_{ 608 }(583, a) \) |
\(1\) | \(1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) |
![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)
