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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4225, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,19]))
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pari:[g,chi] = znchar(Mod(51,4225))
\(\chi_{4225}(51,\cdot)\)
\(\chi_{4225}(376,\cdot)\)
\(\chi_{4225}(701,\cdot)\)
\(\chi_{4225}(1026,\cdot)\)
\(\chi_{4225}(1676,\cdot)\)
\(\chi_{4225}(2001,\cdot)\)
\(\chi_{4225}(2326,\cdot)\)
\(\chi_{4225}(2651,\cdot)\)
\(\chi_{4225}(2976,\cdot)\)
\(\chi_{4225}(3301,\cdot)\)
\(\chi_{4225}(3626,\cdot)\)
\(\chi_{4225}(3951,\cdot)\)
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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 ]
\((677,3551)\) → \((1,e\left(\frac{19}{26}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 4225 }(51, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) |
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sage:chi.jacobi_sum(n)
