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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([8,5]))
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pari:[g,chi] = znchar(Mod(606,4225))
\(\chi_{4225}(606,\cdot)\)
\(\chi_{4225}(746,\cdot)\)
\(\chi_{4225}(1591,\cdot)\)
\(\chi_{4225}(2296,\cdot)\)
\(\chi_{4225}(2436,\cdot)\)
\(\chi_{4225}(3141,\cdot)\)
\(\chi_{4225}(3281,\cdot)\)
\(\chi_{4225}(3986,\cdot)\)
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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 ]
\((677,3551)\) → \((e\left(\frac{2}{5}\right),i)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 4225 }(606, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(-i\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) |
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sage:chi.jacobi_sum(n)
