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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8550, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,1,0]))
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pari:[g,chi] = znchar(Mod(1027,8550))
\(\chi_{8550}(1027,\cdot)\)
\(\chi_{8550}(2053,\cdot)\)
\(\chi_{8550}(2737,\cdot)\)
\(\chi_{8550}(3763,\cdot)\)
\(\chi_{8550}(4447,\cdot)\)
\(\chi_{8550}(5473,\cdot)\)
\(\chi_{8550}(7183,\cdot)\)
\(\chi_{8550}(7867,\cdot)\)
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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 ]
\((1901,1027,1351)\) → \((1,e\left(\frac{1}{20}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 8550 }(1027, a) \) |
\(-1\) | \(1\) | \(i\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(-i\) |
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sage:chi.jacobi_sum(n)
