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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1225, base_ring=CyclotomicField(84))
M = H._module
chi = DirichletCharacter(H, M([63,68]))
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pari:[g,chi] = znchar(Mod(1068,1225))
\(\chi_{1225}(32,\cdot)\)
\(\chi_{1225}(93,\cdot)\)
\(\chi_{1225}(107,\cdot)\)
\(\chi_{1225}(193,\cdot)\)
\(\chi_{1225}(207,\cdot)\)
\(\chi_{1225}(268,\cdot)\)
\(\chi_{1225}(282,\cdot)\)
\(\chi_{1225}(368,\cdot)\)
\(\chi_{1225}(382,\cdot)\)
\(\chi_{1225}(443,\cdot)\)
\(\chi_{1225}(457,\cdot)\)
\(\chi_{1225}(543,\cdot)\)
\(\chi_{1225}(632,\cdot)\)
\(\chi_{1225}(718,\cdot)\)
\(\chi_{1225}(732,\cdot)\)
\(\chi_{1225}(793,\cdot)\)
\(\chi_{1225}(807,\cdot)\)
\(\chi_{1225}(893,\cdot)\)
\(\chi_{1225}(907,\cdot)\)
\(\chi_{1225}(968,\cdot)\)
\(\chi_{1225}(982,\cdot)\)
\(\chi_{1225}(1068,\cdot)\)
\(\chi_{1225}(1082,\cdot)\)
\(\chi_{1225}(1143,\cdot)\)
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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 ]
\((1177,101)\) → \((-i,e\left(\frac{17}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 1225 }(1068, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{4}{21}\right)\) |
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sage:chi.jacobi_sum(n)
