![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(555, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,9,31]))
![Copy content]()
pari:[g,chi] = znchar(Mod(22,555))
\(\chi_{555}(22,\cdot)\)
\(\chi_{555}(163,\cdot)\)
\(\chi_{555}(202,\cdot)\)
\(\chi_{555}(283,\cdot)\)
\(\chi_{555}(298,\cdot)\)
\(\chi_{555}(328,\cdot)\)
\(\chi_{555}(352,\cdot)\)
\(\chi_{555}(388,\cdot)\)
\(\chi_{555}(412,\cdot)\)
\(\chi_{555}(442,\cdot)\)
\(\chi_{555}(457,\cdot)\)
\(\chi_{555}(538,\cdot)\)
![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 ]
\((371,112,76)\) → \((1,i,e\left(\frac{31}{36}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 555 }(22, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{23}{36}\right)\) |
![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)
