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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4225, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([13,5]))
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pari:[g,chi] = znchar(Mod(3817,4225))
\(\chi_{4225}(408,\cdot)\)
\(\chi_{4225}(437,\cdot)\)
\(\chi_{4225}(1253,\cdot)\)
\(\chi_{4225}(2098,\cdot)\)
\(\chi_{4225}(2127,\cdot)\)
\(\chi_{4225}(2972,\cdot)\)
\(\chi_{4225}(3788,\cdot)\)
\(\chi_{4225}(3817,\cdot)\)
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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 ]
\((677,3551)\) → \((e\left(\frac{13}{20}\right),i)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 4225 }(3817, a) \) |
\(1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) |
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sage:chi.jacobi_sum(n)
