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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3600, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,30,10,3]))
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pari:[g,chi] = znchar(Mod(1127,3600))
\(\chi_{3600}(23,\cdot)\)
\(\chi_{3600}(167,\cdot)\)
\(\chi_{3600}(263,\cdot)\)
\(\chi_{3600}(887,\cdot)\)
\(\chi_{3600}(983,\cdot)\)
\(\chi_{3600}(1127,\cdot)\)
\(\chi_{3600}(1463,\cdot)\)
\(\chi_{3600}(1703,\cdot)\)
\(\chi_{3600}(1847,\cdot)\)
\(\chi_{3600}(2183,\cdot)\)
\(\chi_{3600}(2327,\cdot)\)
\(\chi_{3600}(2423,\cdot)\)
\(\chi_{3600}(2567,\cdot)\)
\(\chi_{3600}(2903,\cdot)\)
\(\chi_{3600}(3047,\cdot)\)
\(\chi_{3600}(3287,\cdot)\)
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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 ]
\((3151,901,2801,577)\) → \((-1,-1,e\left(\frac{1}{6}\right),e\left(\frac{1}{20}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3600 }(1127, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{30}\right)\) |
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sage:chi.jacobi_sum(n)
