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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1488, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,15,0,38]))
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pari:[g,chi] = znchar(Mod(229,1488))
\(\chi_{1488}(13,\cdot)\)
\(\chi_{1488}(229,\cdot)\)
\(\chi_{1488}(301,\cdot)\)
\(\chi_{1488}(445,\cdot)\)
\(\chi_{1488}(517,\cdot)\)
\(\chi_{1488}(613,\cdot)\)
\(\chi_{1488}(637,\cdot)\)
\(\chi_{1488}(685,\cdot)\)
\(\chi_{1488}(757,\cdot)\)
\(\chi_{1488}(973,\cdot)\)
\(\chi_{1488}(1045,\cdot)\)
\(\chi_{1488}(1189,\cdot)\)
\(\chi_{1488}(1261,\cdot)\)
\(\chi_{1488}(1357,\cdot)\)
\(\chi_{1488}(1381,\cdot)\)
\(\chi_{1488}(1429,\cdot)\)
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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 ]
\((559,373,497,1057)\) → \((1,i,1,e\left(\frac{19}{30}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(35\) |
| \( \chi_{ 1488 }(229, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) |
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sage:chi.jacobi_sum(n)
