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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2523, base_ring=CyclotomicField(14))
M = H._module
chi = DirichletCharacter(H, M([7,8]))
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pari:[g,chi] = znchar(Mod(605,2523))
\(\chi_{2523}(605,\cdot)\)
\(\chi_{2523}(1031,\cdot)\)
\(\chi_{2523}(1412,\cdot)\)
\(\chi_{2523}(1415,\cdot)\)
\(\chi_{2523}(1619,\cdot)\)
\(\chi_{2523}(2327,\cdot)\)
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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 ]
\((842,1684)\) → \((-1,e\left(\frac{4}{7}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 2523 }(605, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) |
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sage:chi.jacobi_sum(n)
