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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1050, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,9,10]))
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pari:[g,chi] = znchar(Mod(587,1050))
\(\chi_{1050}(83,\cdot)\)
\(\chi_{1050}(167,\cdot)\)
\(\chi_{1050}(377,\cdot)\)
\(\chi_{1050}(503,\cdot)\)
\(\chi_{1050}(587,\cdot)\)
\(\chi_{1050}(713,\cdot)\)
\(\chi_{1050}(797,\cdot)\)
\(\chi_{1050}(923,\cdot)\)
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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 ]
\((701,127,451)\) → \((-1,e\left(\frac{9}{20}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 1050 }(587, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(-i\) |
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sage:chi.jacobi_sum(n)
