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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1309, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,8,15]))
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pari:[g,chi] = znchar(Mod(15,1309))
\(\chi_{1309}(15,\cdot)\)
\(\chi_{1309}(36,\cdot)\)
\(\chi_{1309}(246,\cdot)\)
\(\chi_{1309}(372,\cdot)\)
\(\chi_{1309}(400,\cdot)\)
\(\chi_{1309}(603,\cdot)\)
\(\chi_{1309}(610,\cdot)\)
\(\chi_{1309}(631,\cdot)\)
\(\chi_{1309}(729,\cdot)\)
\(\chi_{1309}(757,\cdot)\)
\(\chi_{1309}(841,\cdot)\)
\(\chi_{1309}(960,\cdot)\)
\(\chi_{1309}(988,\cdot)\)
\(\chi_{1309}(995,\cdot)\)
\(\chi_{1309}(1114,\cdot)\)
\(\chi_{1309}(1226,\cdot)\)
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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 ]
\((1123,596,309)\) → \((1,e\left(\frac{1}{5}\right),e\left(\frac{3}{8}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 1309 }(15, a) \) |
\(1\) | \(1\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{10}\right)\) |
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sage:chi.jacobi_sum(n)
