![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(459, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([0,3]))
![Copy content]()
pari:[g,chi] = znchar(Mod(163,459))
\(\chi_{459}(28,\cdot)\)
\(\chi_{459}(82,\cdot)\)
\(\chi_{459}(109,\cdot)\)
\(\chi_{459}(163,\cdot)\)
\(\chi_{459}(190,\cdot)\)
\(\chi_{459}(244,\cdot)\)
\(\chi_{459}(352,\cdot)\)
\(\chi_{459}(379,\cdot)\)
![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 ]
\((137,190)\) → \((1,e\left(\frac{3}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 459 }(163, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(i\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(-i\) | \(e\left(\frac{11}{16}\right)\) | \(-1\) |
![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)
