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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(608, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,9,28]))
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pari:[g,chi] = znchar(Mod(25,608))
\(\chi_{608}(9,\cdot)\)
\(\chi_{608}(25,\cdot)\)
\(\chi_{608}(73,\cdot)\)
\(\chi_{608}(137,\cdot)\)
\(\chi_{608}(169,\cdot)\)
\(\chi_{608}(233,\cdot)\)
\(\chi_{608}(313,\cdot)\)
\(\chi_{608}(329,\cdot)\)
\(\chi_{608}(377,\cdot)\)
\(\chi_{608}(441,\cdot)\)
\(\chi_{608}(473,\cdot)\)
\(\chi_{608}(537,\cdot)\)
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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 ]
\((191,229,97)\) → \((1,i,e\left(\frac{7}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
| \( \chi_{ 608 }(25, a) \) |
\(1\) | \(1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) |
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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)
