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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(15840, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,5,0,0,36]))
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pari:[g,chi] = znchar(Mod(11611,15840))
\(\chi_{15840}(811,\cdot)\)
\(\chi_{15840}(1531,\cdot)\)
\(\chi_{15840}(2251,\cdot)\)
\(\chi_{15840}(3691,\cdot)\)
\(\chi_{15840}(4771,\cdot)\)
\(\chi_{15840}(5491,\cdot)\)
\(\chi_{15840}(6211,\cdot)\)
\(\chi_{15840}(7651,\cdot)\)
\(\chi_{15840}(8731,\cdot)\)
\(\chi_{15840}(9451,\cdot)\)
\(\chi_{15840}(10171,\cdot)\)
\(\chi_{15840}(11611,\cdot)\)
\(\chi_{15840}(12691,\cdot)\)
\(\chi_{15840}(13411,\cdot)\)
\(\chi_{15840}(14131,\cdot)\)
\(\chi_{15840}(15571,\cdot)\)
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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 ]
\((991,13861,3521,6337,14401)\) → \((-1,e\left(\frac{1}{8}\right),1,1,e\left(\frac{9}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 15840 }(11611, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(i\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) |
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sage:chi.jacobi_sum(n)
