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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1450, base_ring=CyclotomicField(140))
M = H._module
chi = DirichletCharacter(H, M([77,45]))
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pari:[g,chi] = znchar(Mod(773,1450))
\(\chi_{1450}(3,\cdot)\)
\(\chi_{1450}(27,\cdot)\)
\(\chi_{1450}(37,\cdot)\)
\(\chi_{1450}(47,\cdot)\)
\(\chi_{1450}(97,\cdot)\)
\(\chi_{1450}(247,\cdot)\)
\(\chi_{1450}(253,\cdot)\)
\(\chi_{1450}(263,\cdot)\)
\(\chi_{1450}(287,\cdot)\)
\(\chi_{1450}(317,\cdot)\)
\(\chi_{1450}(327,\cdot)\)
\(\chi_{1450}(333,\cdot)\)
\(\chi_{1450}(337,\cdot)\)
\(\chi_{1450}(387,\cdot)\)
\(\chi_{1450}(483,\cdot)\)
\(\chi_{1450}(533,\cdot)\)
\(\chi_{1450}(537,\cdot)\)
\(\chi_{1450}(553,\cdot)\)
\(\chi_{1450}(577,\cdot)\)
\(\chi_{1450}(583,\cdot)\)
\(\chi_{1450}(617,\cdot)\)
\(\chi_{1450}(623,\cdot)\)
\(\chi_{1450}(627,\cdot)\)
\(\chi_{1450}(677,\cdot)\)
\(\chi_{1450}(773,\cdot)\)
\(\chi_{1450}(823,\cdot)\)
\(\chi_{1450}(827,\cdot)\)
\(\chi_{1450}(833,\cdot)\)
\(\chi_{1450}(867,\cdot)\)
\(\chi_{1450}(873,\cdot)\)
...
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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 ]
\((1277,901)\) → \((e\left(\frac{11}{20}\right),e\left(\frac{9}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 1450 }(773, a) \) |
\(1\) | \(1\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{117}{140}\right)\) | \(e\left(\frac{33}{140}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{111}{140}\right)\) | \(e\left(\frac{9}{140}\right)\) | \(e\left(\frac{67}{140}\right)\) | \(e\left(\frac{13}{35}\right)\) |
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sage:chi.jacobi_sum(n)
