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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5200, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,10,3,0]))
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pari:[g,chi] = znchar(Mod(183,5200))
\(\chi_{5200}(183,\cdot)\)
\(\chi_{5200}(1223,\cdot)\)
\(\chi_{5200}(1847,\cdot)\)
\(\chi_{5200}(2263,\cdot)\)
\(\chi_{5200}(2887,\cdot)\)
\(\chi_{5200}(3303,\cdot)\)
\(\chi_{5200}(3927,\cdot)\)
\(\chi_{5200}(4967,\cdot)\)
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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 ]
\((1951,1301,4577,1601)\) → \((-1,-1,e\left(\frac{3}{20}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 5200 }(183, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(i\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) |
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sage:chi.jacobi_sum(n)
