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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(834, base_ring=CyclotomicField(138))
M = H._module
chi = DirichletCharacter(H, M([0,128]))
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pari:[g,chi] = znchar(Mod(169,834))
\(\chi_{834}(7,\cdot)\)
\(\chi_{834}(13,\cdot)\)
\(\chi_{834}(25,\cdot)\)
\(\chi_{834}(31,\cdot)\)
\(\chi_{834}(37,\cdot)\)
\(\chi_{834}(49,\cdot)\)
\(\chi_{834}(67,\cdot)\)
\(\chi_{834}(121,\cdot)\)
\(\chi_{834}(127,\cdot)\)
\(\chi_{834}(163,\cdot)\)
\(\chi_{834}(169,\cdot)\)
\(\chi_{834}(193,\cdot)\)
\(\chi_{834}(205,\cdot)\)
\(\chi_{834}(217,\cdot)\)
\(\chi_{834}(259,\cdot)\)
\(\chi_{834}(283,\cdot)\)
\(\chi_{834}(289,\cdot)\)
\(\chi_{834}(307,\cdot)\)
\(\chi_{834}(313,\cdot)\)
\(\chi_{834}(319,\cdot)\)
\(\chi_{834}(325,\cdot)\)
\(\chi_{834}(349,\cdot)\)
\(\chi_{834}(361,\cdot)\)
\(\chi_{834}(367,\cdot)\)
\(\chi_{834}(385,\cdot)\)
\(\chi_{834}(391,\cdot)\)
\(\chi_{834}(415,\cdot)\)
\(\chi_{834}(421,\cdot)\)
\(\chi_{834}(433,\cdot)\)
\(\chi_{834}(445,\cdot)\)
...
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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 ]
\((557,697)\) → \((1,e\left(\frac{64}{69}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 834 }(169, a) \) |
\(1\) | \(1\) | \(e\left(\frac{53}{69}\right)\) | \(e\left(\frac{26}{69}\right)\) | \(e\left(\frac{34}{69}\right)\) | \(e\left(\frac{25}{69}\right)\) | \(e\left(\frac{17}{69}\right)\) | \(e\left(\frac{40}{69}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{37}{69}\right)\) | \(e\left(\frac{13}{69}\right)\) | \(e\left(\frac{65}{69}\right)\) |
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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)
