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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3150, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,18,20]))
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pari:[g,chi] = znchar(Mod(1621,3150))
\(\chi_{3150}(361,\cdot)\)
\(\chi_{3150}(541,\cdot)\)
\(\chi_{3150}(991,\cdot)\)
\(\chi_{3150}(1171,\cdot)\)
\(\chi_{3150}(1621,\cdot)\)
\(\chi_{3150}(2431,\cdot)\)
\(\chi_{3150}(2881,\cdot)\)
\(\chi_{3150}(3061,\cdot)\)
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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 ]
\((2801,127,451)\) → \((1,e\left(\frac{3}{5}\right),e\left(\frac{2}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 3150 }(1621, a) \) |
\(1\) | \(1\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(1\) |
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sage:chi.jacobi_sum(n)
