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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5200, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,15,12,20]))
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pari:[g,chi] = znchar(Mod(1881,5200))
\(\chi_{5200}(841,\cdot)\)
\(\chi_{5200}(1641,\cdot)\)
\(\chi_{5200}(1881,\cdot)\)
\(\chi_{5200}(2681,\cdot)\)
\(\chi_{5200}(2921,\cdot)\)
\(\chi_{5200}(3721,\cdot)\)
\(\chi_{5200}(3961,\cdot)\)
\(\chi_{5200}(4761,\cdot)\)
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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 ]
\((1951,1301,4577,1601)\) → \((1,-1,e\left(\frac{2}{5}\right),e\left(\frac{2}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 5200 }(1881, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{29}{30}\right)\) |
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sage:chi.jacobi_sum(n)
