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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(34272, base_ring=CyclotomicField(8))
M = H._module
chi = DirichletCharacter(H, M([0,0,4,4,3]))
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pari:[g,chi] = znchar(Mod(7937,34272))
\(\chi_{34272}(1889,\cdot)\)
\(\chi_{34272}(7937,\cdot)\)
\(\chi_{34272}(9953,\cdot)\)
\(\chi_{34272}(34145,\cdot)\)
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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 ]
\((2143,29989,3809,14689,14113)\) → \((1,1,-1,-1,e\left(\frac{3}{8}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 34272 }(7937, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(1\) | \(-i\) | \(e\left(\frac{1}{8}\right)\) | \(-i\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) |
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sage:chi.jacobi_sum(n)
