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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([18,27,22]))
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pari:[g,chi] = znchar(Mod(983,1110))
\(\chi_{1110}(77,\cdot)\)
\(\chi_{1110}(173,\cdot)\)
\(\chi_{1110}(263,\cdot)\)
\(\chi_{1110}(287,\cdot)\)
\(\chi_{1110}(317,\cdot)\)
\(\chi_{1110}(437,\cdot)\)
\(\chi_{1110}(617,\cdot)\)
\(\chi_{1110}(707,\cdot)\)
\(\chi_{1110}(743,\cdot)\)
\(\chi_{1110}(953,\cdot)\)
\(\chi_{1110}(983,\cdot)\)
\(\chi_{1110}(1103,\cdot)\)
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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 ]
\((371,667,631)\) → \((-1,-i,e\left(\frac{11}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(41\) | \(43\) |
| \( \chi_{ 1110 }(983, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(-1\) | \(e\left(\frac{13}{18}\right)\) | \(-i\) |
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sage:chi.jacobi_sum(n)
