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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(695, base_ring=CyclotomicField(138))
M = H._module
chi = DirichletCharacter(H, M([0,52]))
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pari:[g,chi] = znchar(Mod(306,695))
\(\chi_{695}(11,\cdot)\)
\(\chi_{695}(16,\cdot)\)
\(\chi_{695}(31,\cdot)\)
\(\chi_{695}(41,\cdot)\)
\(\chi_{695}(46,\cdot)\)
\(\chi_{695}(51,\cdot)\)
\(\chi_{695}(66,\cdot)\)
\(\chi_{695}(71,\cdot)\)
\(\chi_{695}(81,\cdot)\)
\(\chi_{695}(86,\cdot)\)
\(\chi_{695}(121,\cdot)\)
\(\chi_{695}(136,\cdot)\)
\(\chi_{695}(146,\cdot)\)
\(\chi_{695}(176,\cdot)\)
\(\chi_{695}(186,\cdot)\)
\(\chi_{695}(206,\cdot)\)
\(\chi_{695}(246,\cdot)\)
\(\chi_{695}(256,\cdot)\)
\(\chi_{695}(261,\cdot)\)
\(\chi_{695}(266,\cdot)\)
\(\chi_{695}(276,\cdot)\)
\(\chi_{695}(291,\cdot)\)
\(\chi_{695}(306,\cdot)\)
\(\chi_{695}(316,\cdot)\)
\(\chi_{695}(356,\cdot)\)
\(\chi_{695}(361,\cdot)\)
\(\chi_{695}(391,\cdot)\)
\(\chi_{695}(396,\cdot)\)
\(\chi_{695}(421,\cdot)\)
\(\chi_{695}(426,\cdot)\)
...
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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 ]
\((557,141)\) → \((1,e\left(\frac{26}{69}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 695 }(306, a) \) |
\(1\) | \(1\) | \(e\left(\frac{26}{69}\right)\) | \(e\left(\frac{31}{69}\right)\) | \(e\left(\frac{52}{69}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{58}{69}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{62}{69}\right)\) | \(e\left(\frac{44}{69}\right)\) | \(e\left(\frac{14}{69}\right)\) | \(e\left(\frac{8}{69}\right)\) |
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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)
