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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(309680, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([78,39,117,52,72]))
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pari:[g,chi] = znchar(Mod(31243,309680))
\(\chi_{309680}(67,\cdot)\)
\(\chi_{309680}(31243,\cdot)\)
\(\chi_{309680}(31427,\cdot)\)
\(\chi_{309680}(35163,\cdot)\)
\(\chi_{309680}(50843,\cdot)\)
\(\chi_{309680}(58683,\cdot)\)
\(\chi_{309680}(63963,\cdot)\)
\(\chi_{309680}(67883,\cdot)\)
\(\chi_{309680}(81027,\cdot)\)
\(\chi_{309680}(84947,\cdot)\)
\(\chi_{309680}(90043,\cdot)\)
\(\chi_{309680}(105723,\cdot)\)
\(\chi_{309680}(107083,\cdot)\)
\(\chi_{309680}(111003,\cdot)\)
\(\chi_{309680}(117483,\cdot)\)
\(\chi_{309680}(124147,\cdot)\)
\(\chi_{309680}(126683,\cdot)\)
\(\chi_{309680}(128067,\cdot)\)
\(\chi_{309680}(134523,\cdot)\)
\(\chi_{309680}(137083,\cdot)\)
\(\chi_{309680}(141003,\cdot)\)
\(\chi_{309680}(143747,\cdot)\)
\(\chi_{309680}(151587,\cdot)\)
\(\chi_{309680}(156867,\cdot)\)
\(\chi_{309680}(160787,\cdot)\)
\(\chi_{309680}(165883,\cdot)\)
\(\chi_{309680}(172363,\cdot)\)
\(\chi_{309680}(181563,\cdot)\)
\(\chi_{309680}(182947,\cdot)\)
\(\chi_{309680}(193323,\cdot)\)
...
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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 ]
\((193551,232261,61937,297041,82321)\) → \((-1,i,-i,e\left(\frac{1}{3}\right),e\left(\frac{6}{13}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 309680 }(31243, a) \) |
\(1\) | \(1\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{73}{156}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{121}{156}\right)\) | \(e\left(\frac{29}{156}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{17}{52}\right)\) | \(e\left(\frac{53}{78}\right)\) |
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sage:chi.jacobi_sum(n)
