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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(16830, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([40,20,56,65]))
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pari:[g,chi] = znchar(Mod(4517,16830))
\(\chi_{16830}(107,\cdot)\)
\(\chi_{16830}(503,\cdot)\)
\(\chi_{16830}(2987,\cdot)\)
\(\chi_{16830}(3203,\cdot)\)
\(\chi_{16830}(4193,\cdot)\)
\(\chi_{16830}(4517,\cdot)\)
\(\chi_{16830}(4967,\cdot)\)
\(\chi_{16830}(5777,\cdot)\)
\(\chi_{16830}(6047,\cdot)\)
\(\chi_{16830}(6497,\cdot)\)
\(\chi_{16830}(7163,\cdot)\)
\(\chi_{16830}(7757,\cdot)\)
\(\chi_{16830}(8027,\cdot)\)
\(\chi_{16830}(8153,\cdot)\)
\(\chi_{16830}(9323,\cdot)\)
\(\chi_{16830}(10313,\cdot)\)
\(\chi_{16830}(10853,\cdot)\)
\(\chi_{16830}(11843,\cdot)\)
\(\chi_{16830}(11897,\cdot)\)
\(\chi_{16830}(12383,\cdot)\)
\(\chi_{16830}(13283,\cdot)\)
\(\chi_{16830}(13373,\cdot)\)
\(\chi_{16830}(13427,\cdot)\)
\(\chi_{16830}(13697,\cdot)\)
\(\chi_{16830}(13877,\cdot)\)
\(\chi_{16830}(14273,\cdot)\)
\(\chi_{16830}(14813,\cdot)\)
\(\chi_{16830}(14957,\cdot)\)
\(\chi_{16830}(15407,\cdot)\)
\(\chi_{16830}(15677,\cdot)\)
...
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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 ]
\((7481,3367,1531,8911)\) → \((-1,i,e\left(\frac{7}{10}\right),e\left(\frac{13}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(13\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 16830 }(4517, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{5}\right)\) |
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sage:chi.jacobi_sum(n)
