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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1016, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([0,0,2]))
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pari:[g,chi] = znchar(Mod(9,1016))
\(\chi_{1016}(9,\cdot)\)
\(\chi_{1016}(17,\cdot)\)
\(\chi_{1016}(41,\cdot)\)
\(\chi_{1016}(49,\cdot)\)
\(\chi_{1016}(81,\cdot)\)
\(\chi_{1016}(113,\cdot)\)
\(\chi_{1016}(121,\cdot)\)
\(\chi_{1016}(145,\cdot)\)
\(\chi_{1016}(153,\cdot)\)
\(\chi_{1016}(161,\cdot)\)
\(\chi_{1016}(169,\cdot)\)
\(\chi_{1016}(201,\cdot)\)
\(\chi_{1016}(209,\cdot)\)
\(\chi_{1016}(225,\cdot)\)
\(\chi_{1016}(265,\cdot)\)
\(\chi_{1016}(289,\cdot)\)
\(\chi_{1016}(369,\cdot)\)
\(\chi_{1016}(417,\cdot)\)
\(\chi_{1016}(425,\cdot)\)
\(\chi_{1016}(441,\cdot)\)
\(\chi_{1016}(465,\cdot)\)
\(\chi_{1016}(505,\cdot)\)
\(\chi_{1016}(521,\cdot)\)
\(\chi_{1016}(529,\cdot)\)
\(\chi_{1016}(577,\cdot)\)
\(\chi_{1016}(665,\cdot)\)
\(\chi_{1016}(697,\cdot)\)
\(\chi_{1016}(705,\cdot)\)
\(\chi_{1016}(777,\cdot)\)
\(\chi_{1016}(793,\cdot)\)
...
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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 ]
\((255,509,257)\) → \((1,1,e\left(\frac{1}{63}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 1016 }(9, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{25}{63}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{53}{63}\right)\) |
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sage:chi.jacobi_sum(n)
