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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(693, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([25,0,9]))
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pari:[g,chi] = znchar(Mod(239,693))
\(\chi_{693}(29,\cdot)\)
\(\chi_{693}(50,\cdot)\)
\(\chi_{693}(239,\cdot)\)
\(\chi_{693}(281,\cdot)\)
\(\chi_{693}(365,\cdot)\)
\(\chi_{693}(470,\cdot)\)
\(\chi_{693}(491,\cdot)\)
\(\chi_{693}(596,\cdot)\)
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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 ]
\((155,199,442)\) → \((e\left(\frac{5}{6}\right),1,e\left(\frac{3}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
| \( \chi_{ 693 }(239, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(-1\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{19}{30}\right)\) |
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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)
