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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1060, base_ring=CyclotomicField(52))
M = H._module
chi = DirichletCharacter(H, M([0,39,49]))
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pari:[g,chi] = znchar(Mod(73,1060))
\(\chi_{1060}(73,\cdot)\)
\(\chi_{1060}(157,\cdot)\)
\(\chi_{1060}(177,\cdot)\)
\(\chi_{1060}(217,\cdot)\)
\(\chi_{1060}(253,\cdot)\)
\(\chi_{1060}(277,\cdot)\)
\(\chi_{1060}(313,\cdot)\)
\(\chi_{1060}(353,\cdot)\)
\(\chi_{1060}(373,\cdot)\)
\(\chi_{1060}(457,\cdot)\)
\(\chi_{1060}(533,\cdot)\)
\(\chi_{1060}(557,\cdot)\)
\(\chi_{1060}(597,\cdot)\)
\(\chi_{1060}(617,\cdot)\)
\(\chi_{1060}(657,\cdot)\)
\(\chi_{1060}(697,\cdot)\)
\(\chi_{1060}(773,\cdot)\)
\(\chi_{1060}(817,\cdot)\)
\(\chi_{1060}(893,\cdot)\)
\(\chi_{1060}(933,\cdot)\)
\(\chi_{1060}(973,\cdot)\)
\(\chi_{1060}(993,\cdot)\)
\(\chi_{1060}(1033,\cdot)\)
\(\chi_{1060}(1057,\cdot)\)
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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 ]
\((531,637,161)\) → \((1,-i,e\left(\frac{49}{52}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 1060 }(73, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{49}{52}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{45}{52}\right)\) | \(e\left(\frac{9}{52}\right)\) | \(e\left(\frac{19}{52}\right)\) | \(e\left(\frac{11}{52}\right)\) | \(1\) | \(e\left(\frac{21}{26}\right)\) |
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sage:chi.jacobi_sum(n)
