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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3600, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,15,0,18]))
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pari:[g,chi] = znchar(Mod(19,3600))
\(\chi_{3600}(19,\cdot)\)
\(\chi_{3600}(379,\cdot)\)
\(\chi_{3600}(739,\cdot)\)
\(\chi_{3600}(1459,\cdot)\)
\(\chi_{3600}(1819,\cdot)\)
\(\chi_{3600}(2179,\cdot)\)
\(\chi_{3600}(2539,\cdot)\)
\(\chi_{3600}(3259,\cdot)\)
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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 ]
\((3151,901,2801,577)\) → \((-1,-i,1,e\left(\frac{9}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3600 }(19, a) \) |
\(-1\) | \(1\) | \(-1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) |
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sage:chi.jacobi_sum(n)
