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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8450, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([13,12]))
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pari:[g,chi] = znchar(Mod(1249,8450))
\(\chi_{8450}(599,\cdot)\)
\(\chi_{8450}(1249,\cdot)\)
\(\chi_{8450}(1899,\cdot)\)
\(\chi_{8450}(2549,\cdot)\)
\(\chi_{8450}(3199,\cdot)\)
\(\chi_{8450}(3849,\cdot)\)
\(\chi_{8450}(4499,\cdot)\)
\(\chi_{8450}(5149,\cdot)\)
\(\chi_{8450}(5799,\cdot)\)
\(\chi_{8450}(6449,\cdot)\)
\(\chi_{8450}(7749,\cdot)\)
\(\chi_{8450}(8399,\cdot)\)
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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 ]
\((677,3551)\) → \((-1,e\left(\frac{6}{13}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 8450 }(1249, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(1\) | \(e\left(\frac{8}{13}\right)\) | \(-1\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{6}{13}\right)\) |
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sage:chi.jacobi_sum(n)
