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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4560, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,9,0,9,4]))
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pari:[g,chi] = znchar(Mod(4507,4560))
\(\chi_{4560}(187,\cdot)\)
\(\chi_{4560}(403,\cdot)\)
\(\chi_{4560}(427,\cdot)\)
\(\chi_{4560}(643,\cdot)\)
\(\chi_{4560}(883,\cdot)\)
\(\chi_{4560}(1867,\cdot)\)
\(\chi_{4560}(2107,\cdot)\)
\(\chi_{4560}(2323,\cdot)\)
\(\chi_{4560}(2563,\cdot)\)
\(\chi_{4560}(3787,\cdot)\)
\(\chi_{4560}(4243,\cdot)\)
\(\chi_{4560}(4507,\cdot)\)
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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 ]
\((1711,1141,3041,2737,1921)\) → \((-1,i,1,i,e\left(\frac{1}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 4560 }(4507, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(-1\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{5}{18}\right)\) |
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sage:chi.jacobi_sum(n)
