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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(31360, base_ring=CyclotomicField(112))
M = H._module
chi = DirichletCharacter(H, M([0,63,0,8]))
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pari:[g,chi] = znchar(Mod(14041,31360))
\(\chi_{31360}(41,\cdot)\)
\(\chi_{31360}(601,\cdot)\)
\(\chi_{31360}(1161,\cdot)\)
\(\chi_{31360}(1721,\cdot)\)
\(\chi_{31360}(2281,\cdot)\)
\(\chi_{31360}(3401,\cdot)\)
\(\chi_{31360}(3961,\cdot)\)
\(\chi_{31360}(4521,\cdot)\)
\(\chi_{31360}(5081,\cdot)\)
\(\chi_{31360}(5641,\cdot)\)
\(\chi_{31360}(6201,\cdot)\)
\(\chi_{31360}(7321,\cdot)\)
\(\chi_{31360}(7881,\cdot)\)
\(\chi_{31360}(8441,\cdot)\)
\(\chi_{31360}(9001,\cdot)\)
\(\chi_{31360}(9561,\cdot)\)
\(\chi_{31360}(10121,\cdot)\)
\(\chi_{31360}(11241,\cdot)\)
\(\chi_{31360}(11801,\cdot)\)
\(\chi_{31360}(12361,\cdot)\)
\(\chi_{31360}(12921,\cdot)\)
\(\chi_{31360}(13481,\cdot)\)
\(\chi_{31360}(14041,\cdot)\)
\(\chi_{31360}(15161,\cdot)\)
\(\chi_{31360}(15721,\cdot)\)
\(\chi_{31360}(16281,\cdot)\)
\(\chi_{31360}(16841,\cdot)\)
\(\chi_{31360}(17401,\cdot)\)
\(\chi_{31360}(17961,\cdot)\)
\(\chi_{31360}(19081,\cdot)\)
...
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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 ]
\((17151,28421,18817,10881)\) → \((1,e\left(\frac{9}{16}\right),1,e\left(\frac{1}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 31360 }(14041, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{85}{112}\right)\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{75}{112}\right)\) | \(e\left(\frac{89}{112}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{31}{112}\right)\) | \(e\left(\frac{53}{112}\right)\) | \(1\) |
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sage:chi.jacobi_sum(n)
