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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2523, base_ring=CyclotomicField(812))
M = H._module
chi = DirichletCharacter(H, M([0,303]))
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pari:[g,chi] = znchar(Mod(10,2523))
\(\chi_{2523}(10,\cdot)\)
\(\chi_{2523}(19,\cdot)\)
\(\chi_{2523}(31,\cdot)\)
\(\chi_{2523}(37,\cdot)\)
\(\chi_{2523}(40,\cdot)\)
\(\chi_{2523}(43,\cdot)\)
\(\chi_{2523}(55,\cdot)\)
\(\chi_{2523}(61,\cdot)\)
\(\chi_{2523}(73,\cdot)\)
\(\chi_{2523}(76,\cdot)\)
\(\chi_{2523}(79,\cdot)\)
\(\chi_{2523}(85,\cdot)\)
\(\chi_{2523}(97,\cdot)\)
\(\chi_{2523}(106,\cdot)\)
\(\chi_{2523}(118,\cdot)\)
\(\chi_{2523}(124,\cdot)\)
\(\chi_{2523}(127,\cdot)\)
\(\chi_{2523}(130,\cdot)\)
\(\chi_{2523}(142,\cdot)\)
\(\chi_{2523}(148,\cdot)\)
\(\chi_{2523}(160,\cdot)\)
\(\chi_{2523}(163,\cdot)\)
\(\chi_{2523}(166,\cdot)\)
\(\chi_{2523}(172,\cdot)\)
\(\chi_{2523}(184,\cdot)\)
\(\chi_{2523}(193,\cdot)\)
\(\chi_{2523}(205,\cdot)\)
\(\chi_{2523}(211,\cdot)\)
\(\chi_{2523}(214,\cdot)\)
\(\chi_{2523}(217,\cdot)\)
...
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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 ]
\((842,1684)\) → \((1,e\left(\frac{303}{812}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 2523 }(10, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{303}{812}\right)\) | \(e\left(\frac{303}{406}\right)\) | \(e\left(\frac{281}{406}\right)\) | \(e\left(\frac{62}{203}\right)\) | \(e\left(\frac{97}{812}\right)\) | \(e\left(\frac{53}{812}\right)\) | \(e\left(\frac{183}{812}\right)\) | \(e\left(\frac{11}{406}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{100}{203}\right)\) |
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sage:chi.jacobi_sum(n)
