![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(486, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([38]))
![Copy content]()
pari:[g,chi] = znchar(Mod(469,486))
\(\chi_{486}(19,\cdot)\)
\(\chi_{486}(37,\cdot)\)
\(\chi_{486}(73,\cdot)\)
\(\chi_{486}(91,\cdot)\)
\(\chi_{486}(127,\cdot)\)
\(\chi_{486}(145,\cdot)\)
\(\chi_{486}(181,\cdot)\)
\(\chi_{486}(199,\cdot)\)
\(\chi_{486}(235,\cdot)\)
\(\chi_{486}(253,\cdot)\)
\(\chi_{486}(289,\cdot)\)
\(\chi_{486}(307,\cdot)\)
\(\chi_{486}(343,\cdot)\)
\(\chi_{486}(361,\cdot)\)
\(\chi_{486}(397,\cdot)\)
\(\chi_{486}(415,\cdot)\)
\(\chi_{486}(451,\cdot)\)
\(\chi_{486}(469,\cdot)\)
![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 ]
\(245\) → \(e\left(\frac{19}{27}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 486 }(469, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{2}{27}\right)\) |
![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)
