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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3520, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,5,15,16]))
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pari:[g,chi] = znchar(Mod(1103,3520))
\(\chi_{3520}(47,\cdot)\)
\(\chi_{3520}(367,\cdot)\)
\(\chi_{3520}(687,\cdot)\)
\(\chi_{3520}(1103,\cdot)\)
\(\chi_{3520}(1423,\cdot)\)
\(\chi_{3520}(1743,\cdot)\)
\(\chi_{3520}(1967,\cdot)\)
\(\chi_{3520}(3023,\cdot)\)
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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 ]
\((2751,1541,2817,321)\) → \((-1,i,-i,e\left(\frac{4}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 3520 }(1103, a) \) |
\(1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(i\) | \(i\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{17}{20}\right)\) |
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sage:chi.jacobi_sum(n)
