![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(975, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,21,5]))
![Copy content]()
pari:[g,chi] = znchar(Mod(28,975))
\(\chi_{975}(28,\cdot)\)
\(\chi_{975}(37,\cdot)\)
\(\chi_{975}(58,\cdot)\)
\(\chi_{975}(202,\cdot)\)
\(\chi_{975}(223,\cdot)\)
\(\chi_{975}(253,\cdot)\)
\(\chi_{975}(397,\cdot)\)
\(\chi_{975}(427,\cdot)\)
\(\chi_{975}(448,\cdot)\)
\(\chi_{975}(592,\cdot)\)
\(\chi_{975}(613,\cdot)\)
\(\chi_{975}(622,\cdot)\)
\(\chi_{975}(787,\cdot)\)
\(\chi_{975}(808,\cdot)\)
\(\chi_{975}(817,\cdot)\)
\(\chi_{975}(838,\cdot)\)
![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 ]
\((326,352,301)\) → \((1,e\left(\frac{7}{20}\right),e\left(\frac{1}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
| \( \chi_{ 975 }(28, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{37}{60}\right)\) |
![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)
