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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4225, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([51,40]))
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pari:[g,chi] = znchar(Mod(22,4225))
\(\chi_{4225}(22,\cdot)\)
\(\chi_{4225}(653,\cdot)\)
\(\chi_{4225}(698,\cdot)\)
\(\chi_{4225}(822,\cdot)\)
\(\chi_{4225}(867,\cdot)\)
\(\chi_{4225}(1498,\cdot)\)
\(\chi_{4225}(1667,\cdot)\)
\(\chi_{4225}(1712,\cdot)\)
\(\chi_{4225}(2388,\cdot)\)
\(\chi_{4225}(2512,\cdot)\)
\(\chi_{4225}(3188,\cdot)\)
\(\chi_{4225}(3233,\cdot)\)
\(\chi_{4225}(3402,\cdot)\)
\(\chi_{4225}(4033,\cdot)\)
\(\chi_{4225}(4078,\cdot)\)
\(\chi_{4225}(4202,\cdot)\)
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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 ]
\((677,3551)\) → \((e\left(\frac{17}{20}\right),e\left(\frac{2}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 4225 }(22, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) |
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sage:chi.jacobi_sum(n)
