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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([0,27,7]))
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pari:[g,chi] = znchar(Mod(313,1110))
\(\chi_{1110}(13,\cdot)\)
\(\chi_{1110}(133,\cdot)\)
\(\chi_{1110}(187,\cdot)\)
\(\chi_{1110}(217,\cdot)\)
\(\chi_{1110}(277,\cdot)\)
\(\chi_{1110}(313,\cdot)\)
\(\chi_{1110}(427,\cdot)\)
\(\chi_{1110}(463,\cdot)\)
\(\chi_{1110}(523,\cdot)\)
\(\chi_{1110}(553,\cdot)\)
\(\chi_{1110}(607,\cdot)\)
\(\chi_{1110}(727,\cdot)\)
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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 ]
\((371,667,631)\) → \((1,-i,e\left(\frac{7}{36}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(41\) | \(43\) |
| \( \chi_{ 1110 }(313, a) \) |
\(1\) | \(1\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(-i\) | \(e\left(\frac{7}{18}\right)\) | \(-1\) |
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sage:chi.jacobi_sum(n)
