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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10002, base_ring=CyclotomicField(98))
M = H._module
chi = DirichletCharacter(H, M([0,10]))
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pari:[g,chi] = znchar(Mod(7081,10002))
\(\chi_{10002}(13,\cdot)\)
\(\chi_{10002}(169,\cdot)\)
\(\chi_{10002}(181,\cdot)\)
\(\chi_{10002}(397,\cdot)\)
\(\chi_{10002}(535,\cdot)\)
\(\chi_{10002}(583,\cdot)\)
\(\chi_{10002}(595,\cdot)\)
\(\chi_{10002}(1219,\cdot)\)
\(\chi_{10002}(1405,\cdot)\)
\(\chi_{10002}(1525,\cdot)\)
\(\chi_{10002}(2035,\cdot)\)
\(\chi_{10002}(2197,\cdot)\)
\(\chi_{10002}(2755,\cdot)\)
\(\chi_{10002}(3517,\cdot)\)
\(\chi_{10002}(3631,\cdot)\)
\(\chi_{10002}(3847,\cdot)\)
\(\chi_{10002}(3955,\cdot)\)
\(\chi_{10002}(4255,\cdot)\)
\(\chi_{10002}(5161,\cdot)\)
\(\chi_{10002}(5305,\cdot)\)
\(\chi_{10002}(5665,\cdot)\)
\(\chi_{10002}(5809,\cdot)\)
\(\chi_{10002}(5845,\cdot)\)
\(\chi_{10002}(6169,\cdot)\)
\(\chi_{10002}(6367,\cdot)\)
\(\chi_{10002}(6451,\cdot)\)
\(\chi_{10002}(6817,\cdot)\)
\(\chi_{10002}(7081,\cdot)\)
\(\chi_{10002}(7195,\cdot)\)
\(\chi_{10002}(7399,\cdot)\)
...
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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 ]
\((3335,1669)\) → \((1,e\left(\frac{5}{49}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 10002 }(7081, a) \) |
\(1\) | \(1\) | \(e\left(\frac{33}{49}\right)\) | \(e\left(\frac{10}{49}\right)\) | \(e\left(\frac{8}{49}\right)\) | \(e\left(\frac{8}{49}\right)\) | \(e\left(\frac{24}{49}\right)\) | \(e\left(\frac{10}{49}\right)\) | \(e\left(\frac{5}{49}\right)\) | \(e\left(\frac{17}{49}\right)\) | \(e\left(\frac{11}{49}\right)\) | \(e\left(\frac{23}{49}\right)\) |
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sage:chi.jacobi_sum(n)
