![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(669, base_ring=CyclotomicField(222))
M = H._module
chi = DirichletCharacter(H, M([0,172]))
![Copy content]()
pari:[g,chi] = znchar(Mod(19,669))
\(\chi_{669}(19,\cdot)\)
\(\chi_{669}(25,\cdot)\)
\(\chi_{669}(31,\cdot)\)
\(\chi_{669}(37,\cdot)\)
\(\chi_{669}(43,\cdot)\)
\(\chi_{669}(55,\cdot)\)
\(\chi_{669}(58,\cdot)\)
\(\chi_{669}(73,\cdot)\)
\(\chi_{669}(76,\cdot)\)
\(\chi_{669}(94,\cdot)\)
\(\chi_{669}(100,\cdot)\)
\(\chi_{669}(106,\cdot)\)
\(\chi_{669}(109,\cdot)\)
\(\chi_{669}(121,\cdot)\)
\(\chi_{669}(124,\cdot)\)
\(\chi_{669}(127,\cdot)\)
\(\chi_{669}(130,\cdot)\)
\(\chi_{669}(133,\cdot)\)
\(\chi_{669}(139,\cdot)\)
\(\chi_{669}(148,\cdot)\)
\(\chi_{669}(166,\cdot)\)
\(\chi_{669}(172,\cdot)\)
\(\chi_{669}(175,\cdot)\)
\(\chi_{669}(178,\cdot)\)
\(\chi_{669}(181,\cdot)\)
\(\chi_{669}(199,\cdot)\)
\(\chi_{669}(202,\cdot)\)
\(\chi_{669}(211,\cdot)\)
\(\chi_{669}(217,\cdot)\)
\(\chi_{669}(220,\cdot)\)
...
![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 ]
\((224,226)\) → \((1,e\left(\frac{86}{111}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 669 }(19, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{37}\right)\) | \(e\left(\frac{34}{37}\right)\) | \(e\left(\frac{106}{111}\right)\) | \(e\left(\frac{26}{37}\right)\) | \(e\left(\frac{14}{37}\right)\) | \(e\left(\frac{46}{111}\right)\) | \(e\left(\frac{100}{111}\right)\) | \(e\left(\frac{33}{37}\right)\) | \(e\left(\frac{6}{37}\right)\) | \(e\left(\frac{31}{37}\right)\) |
![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)
