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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([18,18,27,20]))
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pari:[g,chi] = znchar(Mod(5983,6080))
\(\chi_{6080}(1183,\cdot)\)
\(\chi_{6080}(1567,\cdot)\)
\(\chi_{6080}(1887,\cdot)\)
\(\chi_{6080}(2783,\cdot)\)
\(\chi_{6080}(2847,\cdot)\)
\(\chi_{6080}(3103,\cdot)\)
\(\chi_{6080}(4063,\cdot)\)
\(\chi_{6080}(4127,\cdot)\)
\(\chi_{6080}(4767,\cdot)\)
\(\chi_{6080}(5343,\cdot)\)
\(\chi_{6080}(5983,\cdot)\)
\(\chi_{6080}(6047,\cdot)\)
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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 ]
\((191,5701,1217,1921)\) → \((-1,-1,-i,e\left(\frac{5}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 6080 }(5983, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{4}{9}\right)\) |
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sage:chi.jacobi_sum(n)
