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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(820, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,10,33]))
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pari:[g,chi] = znchar(Mod(17,820))
\(\chi_{820}(13,\cdot)\)
\(\chi_{820}(17,\cdot)\)
\(\chi_{820}(53,\cdot)\)
\(\chi_{820}(93,\cdot)\)
\(\chi_{820}(97,\cdot)\)
\(\chi_{820}(117,\cdot)\)
\(\chi_{820}(153,\cdot)\)
\(\chi_{820}(193,\cdot)\)
\(\chi_{820}(233,\cdot)\)
\(\chi_{820}(253,\cdot)\)
\(\chi_{820}(457,\cdot)\)
\(\chi_{820}(477,\cdot)\)
\(\chi_{820}(557,\cdot)\)
\(\chi_{820}(637,\cdot)\)
\(\chi_{820}(757,\cdot)\)
\(\chi_{820}(813,\cdot)\)
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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 ]
\((411,657,621)\) → \((1,i,e\left(\frac{33}{40}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 820 }(17, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(i\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) |
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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)
