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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(508288, base_ring=CyclotomicField(2280))
M = H._module
chi = DirichletCharacter(H, M([0,1425,912,2000]))
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pari:[g,chi] = znchar(Mod(49,508288))
\(\chi_{508288}(49,\cdot)\)
\(\chi_{508288}(273,\cdot)\)
\(\chi_{508288}(1489,\cdot)\)
\(\chi_{508288}(3089,\cdot)\)
\(\chi_{508288}(3921,\cdot)\)
\(\chi_{508288}(4305,\cdot)\)
\(\chi_{508288}(5745,\cdot)\)
\(\chi_{508288}(6737,\cdot)\)
\(\chi_{508288}(6961,\cdot)\)
\(\chi_{508288}(8177,\cdot)\)
\(\chi_{508288}(8561,\cdot)\)
\(\chi_{508288}(9777,\cdot)\)
\(\chi_{508288}(10609,\cdot)\)
\(\chi_{508288}(10993,\cdot)\)
\(\chi_{508288}(12433,\cdot)\)
\(\chi_{508288}(14865,\cdot)\)
\(\chi_{508288}(15249,\cdot)\)
\(\chi_{508288}(16465,\cdot)\)
\(\chi_{508288}(17297,\cdot)\)
\(\chi_{508288}(17681,\cdot)\)
\(\chi_{508288}(19121,\cdot)\)
\(\chi_{508288}(20113,\cdot)\)
\(\chi_{508288}(20337,\cdot)\)
\(\chi_{508288}(21553,\cdot)\)
\(\chi_{508288}(21937,\cdot)\)
\(\chi_{508288}(23153,\cdot)\)
\(\chi_{508288}(23985,\cdot)\)
\(\chi_{508288}(24369,\cdot)\)
\(\chi_{508288}(25809,\cdot)\)
\(\chi_{508288}(26801,\cdot)\)
...
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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 ]
\((166783,174725,323457,14081)\) → \((1,e\left(\frac{5}{8}\right),e\left(\frac{2}{5}\right),e\left(\frac{50}{57}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 508288 }(49, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{2280}\right)\) | \(e\left(\frac{1673}{2280}\right)\) | \(e\left(\frac{239}{380}\right)\) | \(e\left(\frac{11}{1140}\right)\) | \(e\left(\frac{847}{2280}\right)\) | \(e\left(\frac{421}{570}\right)\) | \(e\left(\frac{197}{570}\right)\) | \(e\left(\frac{289}{456}\right)\) | \(e\left(\frac{187}{228}\right)\) | \(e\left(\frac{533}{1140}\right)\) |
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sage:chi.jacobi_sum(n)
