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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3600, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,0,10,9]))
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pari:[g,chi] = znchar(Mod(287,3600))
\(\chi_{3600}(287,\cdot)\)
\(\chi_{3600}(863,\cdot)\)
\(\chi_{3600}(1583,\cdot)\)
\(\chi_{3600}(1727,\cdot)\)
\(\chi_{3600}(2303,\cdot)\)
\(\chi_{3600}(2447,\cdot)\)
\(\chi_{3600}(3023,\cdot)\)
\(\chi_{3600}(3167,\cdot)\)
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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 ]
\((3151,901,2801,577)\) → \((-1,1,-1,e\left(\frac{9}{20}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3600 }(287, a) \) |
\(-1\) | \(1\) | \(-i\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) |
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sage:chi.jacobi_sum(n)
