![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(330, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,5,18]))
![Copy content]()
pari:[g,chi] = znchar(Mod(127,330))
\(\chi_{330}(7,\cdot)\)
\(\chi_{330}(13,\cdot)\)
\(\chi_{330}(73,\cdot)\)
\(\chi_{330}(127,\cdot)\)
\(\chi_{330}(193,\cdot)\)
\(\chi_{330}(217,\cdot)\)
\(\chi_{330}(277,\cdot)\)
\(\chi_{330}(283,\cdot)\)
![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 ]
\((221,67,211)\) → \((1,i,e\left(\frac{9}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 330 }(127, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(-i\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(i\) |
![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)
