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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8925, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,34,0,5]))
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pari:[g,chi] = znchar(Mod(4922,8925))
\(\chi_{8925}(722,\cdot)\)
\(\chi_{8925}(848,\cdot)\)
\(\chi_{8925}(1352,\cdot)\)
\(\chi_{8925}(1583,\cdot)\)
\(\chi_{8925}(2633,\cdot)\)
\(\chi_{8925}(3137,\cdot)\)
\(\chi_{8925}(4292,\cdot)\)
\(\chi_{8925}(4922,\cdot)\)
\(\chi_{8925}(5153,\cdot)\)
\(\chi_{8925}(6077,\cdot)\)
\(\chi_{8925}(6203,\cdot)\)
\(\chi_{8925}(6938,\cdot)\)
\(\chi_{8925}(7862,\cdot)\)
\(\chi_{8925}(7988,\cdot)\)
\(\chi_{8925}(8492,\cdot)\)
\(\chi_{8925}(8723,\cdot)\)
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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 ]
\((5951,6427,2551,8401)\) → \((-1,e\left(\frac{17}{20}\right),1,e\left(\frac{1}{8}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(19\) | \(22\) | \(23\) | \(26\) |
| \( \chi_{ 8925 }(4922, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(-i\) |
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sage:chi.jacobi_sum(n)
