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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6080, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,18,27,2]))
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pari:[g,chi] = znchar(Mod(1313,6080))
\(\chi_{6080}(33,\cdot)\)
\(\chi_{6080}(97,\cdot)\)
\(\chi_{6080}(737,\cdot)\)
\(\chi_{6080}(1313,\cdot)\)
\(\chi_{6080}(1953,\cdot)\)
\(\chi_{6080}(2017,\cdot)\)
\(\chi_{6080}(2977,\cdot)\)
\(\chi_{6080}(3233,\cdot)\)
\(\chi_{6080}(3297,\cdot)\)
\(\chi_{6080}(4193,\cdot)\)
\(\chi_{6080}(4513,\cdot)\)
\(\chi_{6080}(4897,\cdot)\)
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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 ]
\((191,5701,1217,1921)\) → \((1,-1,-i,e\left(\frac{1}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 6080 }(1313, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{17}{18}\right)\) |
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sage:chi.jacobi_sum(n)
