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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(456, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([0,9,0,16]))
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pari:[g,chi] = znchar(Mod(157,456))
\(\chi_{456}(61,\cdot)\)
\(\chi_{456}(85,\cdot)\)
\(\chi_{456}(157,\cdot)\)
\(\chi_{456}(253,\cdot)\)
\(\chi_{456}(301,\cdot)\)
\(\chi_{456}(397,\cdot)\)
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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 ]
\((343,229,305,97)\) → \((1,-1,1,e\left(\frac{8}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 456 }(157, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{18}\right)\) |
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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)
