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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8925, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,12,20,0]))
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pari:[g,chi] = znchar(Mod(256,8925))
\(\chi_{8925}(256,\cdot)\)
\(\chi_{8925}(2041,\cdot)\)
\(\chi_{8925}(3061,\cdot)\)
\(\chi_{8925}(4846,\cdot)\)
\(\chi_{8925}(5611,\cdot)\)
\(\chi_{8925}(6631,\cdot)\)
\(\chi_{8925}(7396,\cdot)\)
\(\chi_{8925}(8416,\cdot)\)
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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 ]
\((5951,6427,2551,8401)\) → \((1,e\left(\frac{2}{5}\right),e\left(\frac{2}{3}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(19\) | \(22\) | \(23\) | \(26\) |
| \( \chi_{ 8925 }(256, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{3}\right)\) |
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sage:chi.jacobi_sum(n)
