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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([0,5,32,0]))
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pari:[g,chi] = znchar(Mod(53,9984))
\(\chi_{9984}(53,\cdot)\)
\(\chi_{9984}(365,\cdot)\)
\(\chi_{9984}(677,\cdot)\)
\(\chi_{9984}(989,\cdot)\)
\(\chi_{9984}(1301,\cdot)\)
\(\chi_{9984}(1613,\cdot)\)
\(\chi_{9984}(1925,\cdot)\)
\(\chi_{9984}(2237,\cdot)\)
\(\chi_{9984}(2549,\cdot)\)
\(\chi_{9984}(2861,\cdot)\)
\(\chi_{9984}(3173,\cdot)\)
\(\chi_{9984}(3485,\cdot)\)
\(\chi_{9984}(3797,\cdot)\)
\(\chi_{9984}(4109,\cdot)\)
\(\chi_{9984}(4421,\cdot)\)
\(\chi_{9984}(4733,\cdot)\)
\(\chi_{9984}(5045,\cdot)\)
\(\chi_{9984}(5357,\cdot)\)
\(\chi_{9984}(5669,\cdot)\)
\(\chi_{9984}(5981,\cdot)\)
\(\chi_{9984}(6293,\cdot)\)
\(\chi_{9984}(6605,\cdot)\)
\(\chi_{9984}(6917,\cdot)\)
\(\chi_{9984}(7229,\cdot)\)
\(\chi_{9984}(7541,\cdot)\)
\(\chi_{9984}(7853,\cdot)\)
\(\chi_{9984}(8165,\cdot)\)
\(\chi_{9984}(8477,\cdot)\)
\(\chi_{9984}(8789,\cdot)\)
\(\chi_{9984}(9101,\cdot)\)
...
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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 ]
\((8191,3589,3329,769)\) → \((1,e\left(\frac{5}{64}\right),-1,1)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 9984 }(53, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{37}{64}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{9}{64}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{51}{64}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{7}{64}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{23}{64}\right)\) |
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sage:chi.jacobi_sum(n)
