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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([10,0,11,10]))
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pari:[g,chi] = znchar(Mod(623,5200))
\(\chi_{5200}(623,\cdot)\)
\(\chi_{5200}(1247,\cdot)\)
\(\chi_{5200}(1663,\cdot)\)
\(\chi_{5200}(2287,\cdot)\)
\(\chi_{5200}(2703,\cdot)\)
\(\chi_{5200}(3327,\cdot)\)
\(\chi_{5200}(4367,\cdot)\)
\(\chi_{5200}(4783,\cdot)\)
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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 ]
\((1951,1301,4577,1601)\) → \((-1,1,e\left(\frac{11}{20}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 5200 }(623, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(-i\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) |
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sage:chi.jacobi_sum(n)
