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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9984, base_ring=CyclotomicField(64))
M = H._module
chi = DirichletCharacter(H, M([32,33,0,16]))
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pari:[g,chi] = znchar(Mod(4987,9984))
\(\chi_{9984}(499,\cdot)\)
\(\chi_{9984}(619,\cdot)\)
\(\chi_{9984}(1123,\cdot)\)
\(\chi_{9984}(1243,\cdot)\)
\(\chi_{9984}(1747,\cdot)\)
\(\chi_{9984}(1867,\cdot)\)
\(\chi_{9984}(2371,\cdot)\)
\(\chi_{9984}(2491,\cdot)\)
\(\chi_{9984}(2995,\cdot)\)
\(\chi_{9984}(3115,\cdot)\)
\(\chi_{9984}(3619,\cdot)\)
\(\chi_{9984}(3739,\cdot)\)
\(\chi_{9984}(4243,\cdot)\)
\(\chi_{9984}(4363,\cdot)\)
\(\chi_{9984}(4867,\cdot)\)
\(\chi_{9984}(4987,\cdot)\)
\(\chi_{9984}(5491,\cdot)\)
\(\chi_{9984}(5611,\cdot)\)
\(\chi_{9984}(6115,\cdot)\)
\(\chi_{9984}(6235,\cdot)\)
\(\chi_{9984}(6739,\cdot)\)
\(\chi_{9984}(6859,\cdot)\)
\(\chi_{9984}(7363,\cdot)\)
\(\chi_{9984}(7483,\cdot)\)
\(\chi_{9984}(7987,\cdot)\)
\(\chi_{9984}(8107,\cdot)\)
\(\chi_{9984}(8611,\cdot)\)
\(\chi_{9984}(8731,\cdot)\)
\(\chi_{9984}(9235,\cdot)\)
\(\chi_{9984}(9355,\cdot)\)
...
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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 ]
\((8191,3589,3329,769)\) → \((-1,e\left(\frac{33}{64}\right),1,i)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 9984 }(4987, a) \) |
\(1\) | \(1\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{39}{64}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{11}{64}\right)\) |
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sage:chi.jacobi_sum(n)
