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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3600, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,0,10,21]))
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pari:[g,chi] = znchar(Mod(1759,3600))
\(\chi_{3600}(79,\cdot)\)
\(\chi_{3600}(319,\cdot)\)
\(\chi_{3600}(1039,\cdot)\)
\(\chi_{3600}(1519,\cdot)\)
\(\chi_{3600}(1759,\cdot)\)
\(\chi_{3600}(2239,\cdot)\)
\(\chi_{3600}(2479,\cdot)\)
\(\chi_{3600}(2959,\cdot)\)
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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 ]
\((3151,901,2801,577)\) → \((-1,1,e\left(\frac{1}{3}\right),e\left(\frac{7}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3600 }(1759, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{15}\right)\) |
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sage:chi.jacobi_sum(n)
