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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7865, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([15,8,5]))
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pari:[g,chi] = znchar(Mod(3778,7865))
\(\chi_{7865}(3518,\cdot)\)
\(\chi_{7865}(3778,\cdot)\)
\(\chi_{7865}(4558,\cdot)\)
\(\chi_{7865}(4607,\cdot)\)
\(\chi_{7865}(4867,\cdot)\)
\(\chi_{7865}(5647,\cdot)\)
\(\chi_{7865}(6053,\cdot)\)
\(\chi_{7865}(7142,\cdot)\)
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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 ]
\((3147,3511,1211)\) → \((-i,e\left(\frac{2}{5}\right),i)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(14\) | \(16\) |
| \( \chi_{ 7865 }(3778, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(i\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) |
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sage:chi.jacobi_sum(n)
