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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1488, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,0,0,2]))
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pari:[g,chi] = znchar(Mod(319,1488))
\(\chi_{1488}(175,\cdot)\)
\(\chi_{1488}(319,\cdot)\)
\(\chi_{1488}(607,\cdot)\)
\(\chi_{1488}(751,\cdot)\)
\(\chi_{1488}(847,\cdot)\)
\(\chi_{1488}(1135,\cdot)\)
\(\chi_{1488}(1423,\cdot)\)
\(\chi_{1488}(1471,\cdot)\)
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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 ]
\((559,373,497,1057)\) → \((-1,1,1,e\left(\frac{1}{15}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(35\) |
| \( \chi_{ 1488 }(319, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{10}\right)\) |
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sage:chi.jacobi_sum(n)
