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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([0,10,7,5]))
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pari:[g,chi] = znchar(Mod(2153,5200))
\(\chi_{5200}(73,\cdot)\)
\(\chi_{5200}(1097,\cdot)\)
\(\chi_{5200}(1113,\cdot)\)
\(\chi_{5200}(2137,\cdot)\)
\(\chi_{5200}(2153,\cdot)\)
\(\chi_{5200}(3177,\cdot)\)
\(\chi_{5200}(4217,\cdot)\)
\(\chi_{5200}(4233,\cdot)\)
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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 ]
\((1951,1301,4577,1601)\) → \((1,-1,e\left(\frac{7}{20}\right),i)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 5200 }(2153, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(-1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) |
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sage:chi.jacobi_sum(n)
