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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([18,29]))
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pari:[g,chi] = znchar(Mod(24,925))
\(\chi_{925}(24,\cdot)\)
\(\chi_{925}(124,\cdot)\)
\(\chi_{925}(224,\cdot)\)
\(\chi_{925}(274,\cdot)\)
\(\chi_{925}(424,\cdot)\)
\(\chi_{925}(449,\cdot)\)
\(\chi_{925}(499,\cdot)\)
\(\chi_{925}(574,\cdot)\)
\(\chi_{925}(624,\cdot)\)
\(\chi_{925}(649,\cdot)\)
\(\chi_{925}(799,\cdot)\)
\(\chi_{925}(849,\cdot)\)
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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 ]
\((852,76)\) → \((-1,e\left(\frac{29}{36}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 925 }(24, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(-i\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{13}{36}\right)\) |
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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)
