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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1110, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,18,5]))
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pari:[g,chi] = znchar(Mod(809,1110))
\(\chi_{1110}(59,\cdot)\)
\(\chi_{1110}(89,\cdot)\)
\(\chi_{1110}(209,\cdot)\)
\(\chi_{1110}(239,\cdot)\)
\(\chi_{1110}(389,\cdot)\)
\(\chi_{1110}(449,\cdot)\)
\(\chi_{1110}(479,\cdot)\)
\(\chi_{1110}(779,\cdot)\)
\(\chi_{1110}(809,\cdot)\)
\(\chi_{1110}(869,\cdot)\)
\(\chi_{1110}(1019,\cdot)\)
\(\chi_{1110}(1049,\cdot)\)
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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 ]
\((371,667,631)\) → \((-1,-1,e\left(\frac{5}{36}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(41\) | \(43\) |
| \( \chi_{ 1110 }(809, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(i\) | \(e\left(\frac{7}{9}\right)\) | \(i\) |
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sage:chi.jacobi_sum(n)
