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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7056, base_ring=CyclotomicField(14))
M = H._module
chi = DirichletCharacter(H, M([7,0,7,4]))
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pari:[g,chi] = znchar(Mod(575,7056))
\(\chi_{7056}(575,\cdot)\)
\(\chi_{7056}(1583,\cdot)\)
\(\chi_{7056}(2591,\cdot)\)
\(\chi_{7056}(3599,\cdot)\)
\(\chi_{7056}(5615,\cdot)\)
\(\chi_{7056}(6623,\cdot)\)
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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 ]
\((6175,1765,785,4609)\) → \((-1,1,-1,e\left(\frac{2}{7}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 7056 }(575, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(-1\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(-1\) | \(e\left(\frac{1}{7}\right)\) |
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sage:chi.jacobi_sum(n)
