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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([0,33,0,0]))
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pari:[g,chi] = znchar(Mod(8581,9984))
\(\chi_{9984}(157,\cdot)\)
\(\chi_{9984}(469,\cdot)\)
\(\chi_{9984}(781,\cdot)\)
\(\chi_{9984}(1093,\cdot)\)
\(\chi_{9984}(1405,\cdot)\)
\(\chi_{9984}(1717,\cdot)\)
\(\chi_{9984}(2029,\cdot)\)
\(\chi_{9984}(2341,\cdot)\)
\(\chi_{9984}(2653,\cdot)\)
\(\chi_{9984}(2965,\cdot)\)
\(\chi_{9984}(3277,\cdot)\)
\(\chi_{9984}(3589,\cdot)\)
\(\chi_{9984}(3901,\cdot)\)
\(\chi_{9984}(4213,\cdot)\)
\(\chi_{9984}(4525,\cdot)\)
\(\chi_{9984}(4837,\cdot)\)
\(\chi_{9984}(5149,\cdot)\)
\(\chi_{9984}(5461,\cdot)\)
\(\chi_{9984}(5773,\cdot)\)
\(\chi_{9984}(6085,\cdot)\)
\(\chi_{9984}(6397,\cdot)\)
\(\chi_{9984}(6709,\cdot)\)
\(\chi_{9984}(7021,\cdot)\)
\(\chi_{9984}(7333,\cdot)\)
\(\chi_{9984}(7645,\cdot)\)
\(\chi_{9984}(7957,\cdot)\)
\(\chi_{9984}(8269,\cdot)\)
\(\chi_{9984}(8581,\cdot)\)
\(\chi_{9984}(8893,\cdot)\)
\(\chi_{9984}(9205,\cdot)\)
...
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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 ]
\((8191,3589,3329,769)\) → \((1,e\left(\frac{33}{64}\right),1,1)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 9984 }(8581, a) \) |
\(1\) | \(1\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{53}{64}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{43}{64}\right)\) |
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sage:chi.jacobi_sum(n)
