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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2312, base_ring=CyclotomicField(272))
M = H._module
chi = DirichletCharacter(H, M([136,0,9]))
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pari:[g,chi] = znchar(Mod(31,2312))
\(\chi_{2312}(7,\cdot)\)
\(\chi_{2312}(23,\cdot)\)
\(\chi_{2312}(31,\cdot)\)
\(\chi_{2312}(39,\cdot)\)
\(\chi_{2312}(63,\cdot)\)
\(\chi_{2312}(71,\cdot)\)
\(\chi_{2312}(79,\cdot)\)
\(\chi_{2312}(95,\cdot)\)
\(\chi_{2312}(143,\cdot)\)
\(\chi_{2312}(159,\cdot)\)
\(\chi_{2312}(167,\cdot)\)
\(\chi_{2312}(175,\cdot)\)
\(\chi_{2312}(199,\cdot)\)
\(\chi_{2312}(207,\cdot)\)
\(\chi_{2312}(215,\cdot)\)
\(\chi_{2312}(231,\cdot)\)
\(\chi_{2312}(279,\cdot)\)
\(\chi_{2312}(295,\cdot)\)
\(\chi_{2312}(303,\cdot)\)
\(\chi_{2312}(311,\cdot)\)
\(\chi_{2312}(335,\cdot)\)
\(\chi_{2312}(343,\cdot)\)
\(\chi_{2312}(351,\cdot)\)
\(\chi_{2312}(367,\cdot)\)
\(\chi_{2312}(415,\cdot)\)
\(\chi_{2312}(431,\cdot)\)
\(\chi_{2312}(439,\cdot)\)
\(\chi_{2312}(471,\cdot)\)
\(\chi_{2312}(479,\cdot)\)
\(\chi_{2312}(487,\cdot)\)
...
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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 ]
\((1735,1157,1737)\) → \((-1,1,e\left(\frac{9}{272}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 2312 }(31, a) \) |
\(1\) | \(1\) | \(e\left(\frac{145}{272}\right)\) | \(e\left(\frac{157}{272}\right)\) | \(e\left(\frac{171}{272}\right)\) | \(e\left(\frac{9}{136}\right)\) | \(e\left(\frac{71}{272}\right)\) | \(e\left(\frac{33}{68}\right)\) | \(e\left(\frac{15}{136}\right)\) | \(e\left(\frac{131}{136}\right)\) | \(e\left(\frac{11}{68}\right)\) | \(e\left(\frac{239}{272}\right)\) |
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sage:chi.jacobi_sum(n)
