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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(512, base_ring=CyclotomicField(64))
M = H._module
chi = DirichletCharacter(H, M([0,63]))
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pari:[g,chi] = znchar(Mod(41,512))
\(\chi_{512}(9,\cdot)\)
\(\chi_{512}(25,\cdot)\)
\(\chi_{512}(41,\cdot)\)
\(\chi_{512}(57,\cdot)\)
\(\chi_{512}(73,\cdot)\)
\(\chi_{512}(89,\cdot)\)
\(\chi_{512}(105,\cdot)\)
\(\chi_{512}(121,\cdot)\)
\(\chi_{512}(137,\cdot)\)
\(\chi_{512}(153,\cdot)\)
\(\chi_{512}(169,\cdot)\)
\(\chi_{512}(185,\cdot)\)
\(\chi_{512}(201,\cdot)\)
\(\chi_{512}(217,\cdot)\)
\(\chi_{512}(233,\cdot)\)
\(\chi_{512}(249,\cdot)\)
\(\chi_{512}(265,\cdot)\)
\(\chi_{512}(281,\cdot)\)
\(\chi_{512}(297,\cdot)\)
\(\chi_{512}(313,\cdot)\)
\(\chi_{512}(329,\cdot)\)
\(\chi_{512}(345,\cdot)\)
\(\chi_{512}(361,\cdot)\)
\(\chi_{512}(377,\cdot)\)
\(\chi_{512}(393,\cdot)\)
\(\chi_{512}(409,\cdot)\)
\(\chi_{512}(425,\cdot)\)
\(\chi_{512}(441,\cdot)\)
\(\chi_{512}(457,\cdot)\)
\(\chi_{512}(473,\cdot)\)
...
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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 ]
\((511,5)\) → \((1,e\left(\frac{63}{64}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 512 }(41, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{64}\right)\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{43}{64}\right)\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{41}{64}\right)\) | \(e\left(\frac{19}{64}\right)\) |
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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)
