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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([0,137]))
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pari:[g,chi] = znchar(Mod(487,834))
\(\chi_{834}(19,\cdot)\)
\(\chi_{834}(61,\cdot)\)
\(\chi_{834}(73,\cdot)\)
\(\chi_{834}(85,\cdot)\)
\(\chi_{834}(109,\cdot)\)
\(\chi_{834}(115,\cdot)\)
\(\chi_{834}(151,\cdot)\)
\(\chi_{834}(157,\cdot)\)
\(\chi_{834}(211,\cdot)\)
\(\chi_{834}(229,\cdot)\)
\(\chi_{834}(241,\cdot)\)
\(\chi_{834}(247,\cdot)\)
\(\chi_{834}(253,\cdot)\)
\(\chi_{834}(265,\cdot)\)
\(\chi_{834}(271,\cdot)\)
\(\chi_{834}(295,\cdot)\)
\(\chi_{834}(331,\cdot)\)
\(\chi_{834}(379,\cdot)\)
\(\chi_{834}(397,\cdot)\)
\(\chi_{834}(439,\cdot)\)
\(\chi_{834}(457,\cdot)\)
\(\chi_{834}(475,\cdot)\)
\(\chi_{834}(487,\cdot)\)
\(\chi_{834}(505,\cdot)\)
\(\chi_{834}(547,\cdot)\)
\(\chi_{834}(559,\cdot)\)
\(\chi_{834}(571,\cdot)\)
\(\chi_{834}(577,\cdot)\)
\(\chi_{834}(649,\cdot)\)
\(\chi_{834}(667,\cdot)\)
...
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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 ]
\((557,697)\) → \((1,e\left(\frac{137}{138}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 834 }(487, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{26}{69}\right)\) | \(e\left(\frac{44}{69}\right)\) | \(e\left(\frac{31}{69}\right)\) | \(e\left(\frac{37}{69}\right)\) | \(e\left(\frac{31}{138}\right)\) | \(e\left(\frac{77}{138}\right)\) | \(e\left(\frac{37}{46}\right)\) | \(e\left(\frac{52}{69}\right)\) | \(e\left(\frac{22}{69}\right)\) | \(e\left(\frac{41}{69}\right)\) |
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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)
