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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([0,13,10]))
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pari:[g,chi] = znchar(Mod(17,1060))
\(\chi_{1060}(17,\cdot)\)
\(\chi_{1060}(37,\cdot)\)
\(\chi_{1060}(57,\cdot)\)
\(\chi_{1060}(93,\cdot)\)
\(\chi_{1060}(113,\cdot)\)
\(\chi_{1060}(117,\cdot)\)
\(\chi_{1060}(197,\cdot)\)
\(\chi_{1060}(237,\cdot)\)
\(\chi_{1060}(377,\cdot)\)
\(\chi_{1060}(433,\cdot)\)
\(\chi_{1060}(453,\cdot)\)
\(\chi_{1060}(517,\cdot)\)
\(\chi_{1060}(537,\cdot)\)
\(\chi_{1060}(573,\cdot)\)
\(\chi_{1060}(653,\cdot)\)
\(\chi_{1060}(673,\cdot)\)
\(\chi_{1060}(693,\cdot)\)
\(\chi_{1060}(753,\cdot)\)
\(\chi_{1060}(833,\cdot)\)
\(\chi_{1060}(857,\cdot)\)
\(\chi_{1060}(873,\cdot)\)
\(\chi_{1060}(877,\cdot)\)
\(\chi_{1060}(997,\cdot)\)
\(\chi_{1060}(1013,\cdot)\)
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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 ]
\((531,637,161)\) → \((1,i,e\left(\frac{5}{26}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 1060 }(17, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{52}\right)\) | \(e\left(\frac{49}{52}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{19}{52}\right)\) | \(e\left(\frac{9}{52}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(i\) | \(e\left(\frac{3}{52}\right)\) |
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sage:chi.jacobi_sum(n)
