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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1309, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([20,21,0]))
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pari:[g,chi] = znchar(Mod(18,1309))
\(\chi_{1309}(18,\cdot)\)
\(\chi_{1309}(205,\cdot)\)
\(\chi_{1309}(613,\cdot)\)
\(\chi_{1309}(732,\cdot)\)
\(\chi_{1309}(800,\cdot)\)
\(\chi_{1309}(919,\cdot)\)
\(\chi_{1309}(970,\cdot)\)
\(\chi_{1309}(1157,\cdot)\)
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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 ]
\((1123,596,309)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{7}{10}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 1309 }(18, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{10}\right)\) |
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sage:chi.jacobi_sum(n)
