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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8464, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,11,24]))
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pari:[g,chi] = znchar(Mod(501,8464))
\(\chi_{8464}(501,\cdot)\)
\(\chi_{8464}(1757,\cdot)\)
\(\chi_{8464}(1853,\cdot)\)
\(\chi_{8464}(2053,\cdot)\)
\(\chi_{8464}(2293,\cdot)\)
\(\chi_{8464}(3429,\cdot)\)
\(\chi_{8464}(3573,\cdot)\)
\(\chi_{8464}(3661,\cdot)\)
\(\chi_{8464}(3821,\cdot)\)
\(\chi_{8464}(4037,\cdot)\)
\(\chi_{8464}(4733,\cdot)\)
\(\chi_{8464}(5989,\cdot)\)
\(\chi_{8464}(6085,\cdot)\)
\(\chi_{8464}(6285,\cdot)\)
\(\chi_{8464}(6525,\cdot)\)
\(\chi_{8464}(7661,\cdot)\)
\(\chi_{8464}(7805,\cdot)\)
\(\chi_{8464}(7893,\cdot)\)
\(\chi_{8464}(8053,\cdot)\)
\(\chi_{8464}(8269,\cdot)\)
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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 ]
\((7407,2117,6353)\) → \((1,i,e\left(\frac{6}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 8464 }(501, a) \) |
\(1\) | \(1\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{15}{44}\right)\) |
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sage:chi.jacobi_sum(n)
