![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(357, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,32,39]))
![Copy content]()
pari:[g,chi] = znchar(Mod(46,357))
\(\chi_{357}(37,\cdot)\)
\(\chi_{357}(46,\cdot)\)
\(\chi_{357}(58,\cdot)\)
\(\chi_{357}(79,\cdot)\)
\(\chi_{357}(88,\cdot)\)
\(\chi_{357}(109,\cdot)\)
\(\chi_{357}(130,\cdot)\)
\(\chi_{357}(142,\cdot)\)
\(\chi_{357}(163,\cdot)\)
\(\chi_{357}(184,\cdot)\)
\(\chi_{357}(193,\cdot)\)
\(\chi_{357}(214,\cdot)\)
\(\chi_{357}(226,\cdot)\)
\(\chi_{357}(235,\cdot)\)
\(\chi_{357}(277,\cdot)\)
\(\chi_{357}(352,\cdot)\)
![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 ]
\((239,52,190)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{13}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(19\) | \(20\) |
| \( \chi_{ 357 }(46, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(i\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{13}{16}\right)\) |
![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)
