![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(939, base_ring=CyclotomicField(312))
M = H._module
chi = DirichletCharacter(H, M([0,161]))
![Copy content]()
pari:[g,chi] = znchar(Mod(160,939))
\(\chi_{939}(10,\cdot)\)
\(\chi_{939}(28,\cdot)\)
\(\chi_{939}(31,\cdot)\)
\(\chi_{939}(34,\cdot)\)
\(\chi_{939}(37,\cdot)\)
\(\chi_{939}(46,\cdot)\)
\(\chi_{939}(55,\cdot)\)
\(\chi_{939}(67,\cdot)\)
\(\chi_{939}(91,\cdot)\)
\(\chi_{939}(94,\cdot)\)
\(\chi_{939}(106,\cdot)\)
\(\chi_{939}(109,\cdot)\)
\(\chi_{939}(112,\cdot)\)
\(\chi_{939}(127,\cdot)\)
\(\chi_{939}(130,\cdot)\)
\(\chi_{939}(154,\cdot)\)
\(\chi_{939}(160,\cdot)\)
\(\chi_{939}(187,\cdot)\)
\(\chi_{939}(190,\cdot)\)
\(\chi_{939}(193,\cdot)\)
\(\chi_{939}(211,\cdot)\)
\(\chi_{939}(223,\cdot)\)
\(\chi_{939}(229,\cdot)\)
\(\chi_{939}(244,\cdot)\)
\(\chi_{939}(250,\cdot)\)
\(\chi_{939}(253,\cdot)\)
\(\chi_{939}(268,\cdot)\)
\(\chi_{939}(292,\cdot)\)
\(\chi_{939}(298,\cdot)\)
\(\chi_{939}(328,\cdot)\)
...
![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 ]
\((314,10)\) → \((1,e\left(\frac{161}{312}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 939 }(160, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{61}{156}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{49}{104}\right)\) | \(e\left(\frac{9}{52}\right)\) | \(e\left(\frac{161}{312}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{67}{156}\right)\) | \(e\left(\frac{269}{312}\right)\) | \(e\left(\frac{22}{39}\right)\) |
![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)
