![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(555, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,9,32]))
![Copy content]()
pari:[g,chi] = znchar(Mod(7,555))
\(\chi_{555}(7,\cdot)\)
\(\chi_{555}(118,\cdot)\)
\(\chi_{555}(127,\cdot)\)
\(\chi_{555}(157,\cdot)\)
\(\chi_{555}(238,\cdot)\)
\(\chi_{555}(268,\cdot)\)
\(\chi_{555}(292,\cdot)\)
\(\chi_{555}(367,\cdot)\)
\(\chi_{555}(382,\cdot)\)
\(\chi_{555}(403,\cdot)\)
\(\chi_{555}(478,\cdot)\)
\(\chi_{555}(493,\cdot)\)
![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 ]
\((371,112,76)\) → \((1,i,e\left(\frac{8}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 555 }(7, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) |
![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)
