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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1016, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,0,16]))
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pari:[g,chi] = znchar(Mod(25,1016))
\(\chi_{1016}(25,\cdot)\)
\(\chi_{1016}(73,\cdot)\)
\(\chi_{1016}(177,\cdot)\)
\(\chi_{1016}(249,\cdot)\)
\(\chi_{1016}(457,\cdot)\)
\(\chi_{1016}(481,\cdot)\)
\(\chi_{1016}(569,\cdot)\)
\(\chi_{1016}(625,\cdot)\)
\(\chi_{1016}(673,\cdot)\)
\(\chi_{1016}(729,\cdot)\)
\(\chi_{1016}(809,\cdot)\)
\(\chi_{1016}(849,\cdot)\)
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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 ]
\((255,509,257)\) → \((1,1,e\left(\frac{8}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 1016 }(25, a) \) |
\(1\) | \(1\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(1\) | \(e\left(\frac{4}{21}\right)\) |
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sage:chi.jacobi_sum(n)
