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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5200, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,30,27,50]))
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pari:[g,chi] = znchar(Mod(1687,5200))
\(\chi_{5200}(23,\cdot)\)
\(\chi_{5200}(647,\cdot)\)
\(\chi_{5200}(823,\cdot)\)
\(\chi_{5200}(1063,\cdot)\)
\(\chi_{5200}(1447,\cdot)\)
\(\chi_{5200}(1687,\cdot)\)
\(\chi_{5200}(1863,\cdot)\)
\(\chi_{5200}(2103,\cdot)\)
\(\chi_{5200}(2487,\cdot)\)
\(\chi_{5200}(2727,\cdot)\)
\(\chi_{5200}(2903,\cdot)\)
\(\chi_{5200}(3527,\cdot)\)
\(\chi_{5200}(3767,\cdot)\)
\(\chi_{5200}(4183,\cdot)\)
\(\chi_{5200}(4567,\cdot)\)
\(\chi_{5200}(4983,\cdot)\)
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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 ]
\((1951,1301,4577,1601)\) → \((-1,-1,e\left(\frac{9}{20}\right),e\left(\frac{5}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 5200 }(1687, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{11}{15}\right)\) |
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sage:chi.jacobi_sum(n)
