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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4998, base_ring=CyclotomicField(112))
M = H._module
chi = DirichletCharacter(H, M([0,104,49]))
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pari:[g,chi] = znchar(Mod(2323,4998))
\(\chi_{4998}(139,\cdot)\)
\(\chi_{4998}(181,\cdot)\)
\(\chi_{4998}(265,\cdot)\)
\(\chi_{4998}(517,\cdot)\)
\(\chi_{4998}(601,\cdot)\)
\(\chi_{4998}(643,\cdot)\)
\(\chi_{4998}(811,\cdot)\)
\(\chi_{4998}(853,\cdot)\)
\(\chi_{4998}(895,\cdot)\)
\(\chi_{4998}(1231,\cdot)\)
\(\chi_{4998}(1315,\cdot)\)
\(\chi_{4998}(1357,\cdot)\)
\(\chi_{4998}(1399,\cdot)\)
\(\chi_{4998}(1525,\cdot)\)
\(\chi_{4998}(1609,\cdot)\)
\(\chi_{4998}(1693,\cdot)\)
\(\chi_{4998}(1945,\cdot)\)
\(\chi_{4998}(2029,\cdot)\)
\(\chi_{4998}(2071,\cdot)\)
\(\chi_{4998}(2113,\cdot)\)
\(\chi_{4998}(2239,\cdot)\)
\(\chi_{4998}(2281,\cdot)\)
\(\chi_{4998}(2323,\cdot)\)
\(\chi_{4998}(2407,\cdot)\)
\(\chi_{4998}(2659,\cdot)\)
\(\chi_{4998}(2785,\cdot)\)
\(\chi_{4998}(2827,\cdot)\)
\(\chi_{4998}(2953,\cdot)\)
\(\chi_{4998}(2995,\cdot)\)
\(\chi_{4998}(3121,\cdot)\)
...
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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 ]
\((1667,2551,4117)\) → \((1,e\left(\frac{13}{14}\right),e\left(\frac{7}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 4998 }(2323, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{112}\right)\) | \(e\left(\frac{23}{112}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{95}{112}\right)\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{45}{112}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{17}{112}\right)\) | \(e\left(\frac{83}{112}\right)\) |
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sage:chi.jacobi_sum(n)
