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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3751, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([18,10]))
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pari:[g,chi] = znchar(Mod(614,3751))
\(\chi_{3751}(366,\cdot)\)
\(\chi_{3751}(614,\cdot)\)
\(\chi_{3751}(1358,\cdot)\)
\(\chi_{3751}(1896,\cdot)\)
\(\chi_{3751}(2423,\cdot)\)
\(\chi_{3751}(2671,\cdot)\)
\(\chi_{3751}(3415,\cdot)\)
\(\chi_{3751}(3590,\cdot)\)
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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 ]
\((2543,2421)\) → \((e\left(\frac{3}{5}\right),e\left(\frac{1}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 3751 }(614, a) \) |
\(1\) | \(1\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) |
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sage:chi.jacobi_sum(n)
