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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(507, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([0,55]))
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pari:[g,chi] = znchar(Mod(394,507))
\(\chi_{507}(4,\cdot)\)
\(\chi_{507}(10,\cdot)\)
\(\chi_{507}(43,\cdot)\)
\(\chi_{507}(49,\cdot)\)
\(\chi_{507}(82,\cdot)\)
\(\chi_{507}(88,\cdot)\)
\(\chi_{507}(121,\cdot)\)
\(\chi_{507}(127,\cdot)\)
\(\chi_{507}(160,\cdot)\)
\(\chi_{507}(166,\cdot)\)
\(\chi_{507}(199,\cdot)\)
\(\chi_{507}(205,\cdot)\)
\(\chi_{507}(238,\cdot)\)
\(\chi_{507}(244,\cdot)\)
\(\chi_{507}(277,\cdot)\)
\(\chi_{507}(283,\cdot)\)
\(\chi_{507}(322,\cdot)\)
\(\chi_{507}(355,\cdot)\)
\(\chi_{507}(394,\cdot)\)
\(\chi_{507}(400,\cdot)\)
\(\chi_{507}(433,\cdot)\)
\(\chi_{507}(439,\cdot)\)
\(\chi_{507}(472,\cdot)\)
\(\chi_{507}(478,\cdot)\)
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sage:chi.galois_orbit()
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pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((170,340)\) → \((1,e\left(\frac{55}{78}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 507 }(394, a) \) |
\(1\) | \(1\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{49}{78}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{37}{39}\right)\) |
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sage:chi.jacobi_sum(n)
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sage:chi.gauss_sum(a)
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pari:znchargauss(g,chi,a)
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sage:chi.jacobi_sum(n)
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sage:chi.kloosterman_sum(a,b)
