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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(925, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([27,17]))
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pari:[g,chi] = znchar(Mod(18,925))
\(\chi_{925}(18,\cdot)\)
\(\chi_{925}(143,\cdot)\)
\(\chi_{925}(168,\cdot)\)
\(\chi_{925}(207,\cdot)\)
\(\chi_{925}(257,\cdot)\)
\(\chi_{925}(457,\cdot)\)
\(\chi_{925}(468,\cdot)\)
\(\chi_{925}(668,\cdot)\)
\(\chi_{925}(718,\cdot)\)
\(\chi_{925}(757,\cdot)\)
\(\chi_{925}(782,\cdot)\)
\(\chi_{925}(907,\cdot)\)
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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 ]
\((852,76)\) → \((-i,e\left(\frac{17}{36}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 925 }(18, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(-i\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{4}{9}\right)\) |
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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)
