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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6800, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([40,40,72,5]))
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pari:[g,chi] = znchar(Mod(2519,6800))
\(\chi_{6800}(39,\cdot)\)
\(\chi_{6800}(279,\cdot)\)
\(\chi_{6800}(439,\cdot)\)
\(\chi_{6800}(759,\cdot)\)
\(\chi_{6800}(839,\cdot)\)
\(\chi_{6800}(1159,\cdot)\)
\(\chi_{6800}(1319,\cdot)\)
\(\chi_{6800}(1559,\cdot)\)
\(\chi_{6800}(1639,\cdot)\)
\(\chi_{6800}(2119,\cdot)\)
\(\chi_{6800}(2519,\cdot)\)
\(\chi_{6800}(2679,\cdot)\)
\(\chi_{6800}(2759,\cdot)\)
\(\chi_{6800}(2919,\cdot)\)
\(\chi_{6800}(3159,\cdot)\)
\(\chi_{6800}(3479,\cdot)\)
\(\chi_{6800}(3559,\cdot)\)
\(\chi_{6800}(3879,\cdot)\)
\(\chi_{6800}(4039,\cdot)\)
\(\chi_{6800}(4119,\cdot)\)
\(\chi_{6800}(4279,\cdot)\)
\(\chi_{6800}(4359,\cdot)\)
\(\chi_{6800}(4519,\cdot)\)
\(\chi_{6800}(4839,\cdot)\)
\(\chi_{6800}(4919,\cdot)\)
\(\chi_{6800}(5239,\cdot)\)
\(\chi_{6800}(5479,\cdot)\)
\(\chi_{6800}(5639,\cdot)\)
\(\chi_{6800}(5719,\cdot)\)
\(\chi_{6800}(5879,\cdot)\)
...
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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 ]
\((5951,1701,2177,1601)\) → \((-1,-1,e\left(\frac{9}{10}\right),e\left(\frac{1}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 6800 }(2519, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{9}{80}\right)\) |
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sage:chi.jacobi_sum(n)
