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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9800, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,0,21,5]))
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pari:[g,chi] = znchar(Mod(3559,9800))
\(\chi_{9800}(2959,\cdot)\)
\(\chi_{9800}(3559,\cdot)\)
\(\chi_{9800}(4919,\cdot)\)
\(\chi_{9800}(5519,\cdot)\)
\(\chi_{9800}(6879,\cdot)\)
\(\chi_{9800}(7479,\cdot)\)
\(\chi_{9800}(8839,\cdot)\)
\(\chi_{9800}(9439,\cdot)\)
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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 ]
\((7351,4901,1177,5001)\) → \((-1,1,e\left(\frac{7}{10}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 9800 }(3559, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{4}{15}\right)\) |
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sage:chi.jacobi_sum(n)
