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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(834, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([23,8]))
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pari:[g,chi] = znchar(Mod(611,834))
\(\chi_{834}(65,\cdot)\)
\(\chi_{834}(77,\cdot)\)
\(\chi_{834}(125,\cdot)\)
\(\chi_{834}(131,\cdot)\)
\(\chi_{834}(173,\cdot)\)
\(\chi_{834}(191,\cdot)\)
\(\chi_{834}(203,\cdot)\)
\(\chi_{834}(239,\cdot)\)
\(\chi_{834}(245,\cdot)\)
\(\chi_{834}(251,\cdot)\)
\(\chi_{834}(323,\cdot)\)
\(\chi_{834}(335,\cdot)\)
\(\chi_{834}(341,\cdot)\)
\(\chi_{834}(407,\cdot)\)
\(\chi_{834}(461,\cdot)\)
\(\chi_{834}(497,\cdot)\)
\(\chi_{834}(533,\cdot)\)
\(\chi_{834}(611,\cdot)\)
\(\chi_{834}(635,\cdot)\)
\(\chi_{834}(647,\cdot)\)
\(\chi_{834}(701,\cdot)\)
\(\chi_{834}(731,\cdot)\)
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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 ]
\((557,697)\) → \((-1,e\left(\frac{4}{23}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 834 }(611, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{21}{46}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{33}{46}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{5}{46}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{9}{46}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{39}{46}\right)\) | \(e\left(\frac{17}{23}\right)\) |
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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)
