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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(768, base_ring=CyclotomicField(64))
M = H._module
chi = DirichletCharacter(H, M([0,57,0]))
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pari:[g,chi] = znchar(Mod(421,768))
\(\chi_{768}(13,\cdot)\)
\(\chi_{768}(37,\cdot)\)
\(\chi_{768}(61,\cdot)\)
\(\chi_{768}(85,\cdot)\)
\(\chi_{768}(109,\cdot)\)
\(\chi_{768}(133,\cdot)\)
\(\chi_{768}(157,\cdot)\)
\(\chi_{768}(181,\cdot)\)
\(\chi_{768}(205,\cdot)\)
\(\chi_{768}(229,\cdot)\)
\(\chi_{768}(253,\cdot)\)
\(\chi_{768}(277,\cdot)\)
\(\chi_{768}(301,\cdot)\)
\(\chi_{768}(325,\cdot)\)
\(\chi_{768}(349,\cdot)\)
\(\chi_{768}(373,\cdot)\)
\(\chi_{768}(397,\cdot)\)
\(\chi_{768}(421,\cdot)\)
\(\chi_{768}(445,\cdot)\)
\(\chi_{768}(469,\cdot)\)
\(\chi_{768}(493,\cdot)\)
\(\chi_{768}(517,\cdot)\)
\(\chi_{768}(541,\cdot)\)
\(\chi_{768}(565,\cdot)\)
\(\chi_{768}(589,\cdot)\)
\(\chi_{768}(613,\cdot)\)
\(\chi_{768}(637,\cdot)\)
\(\chi_{768}(661,\cdot)\)
\(\chi_{768}(685,\cdot)\)
\(\chi_{768}(709,\cdot)\)
...
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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 ]
\((511,517,257)\) → \((1,e\left(\frac{57}{64}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 768 }(421, a) \) |
\(1\) | \(1\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{45}{64}\right)\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{31}{64}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{35}{64}\right)\) | \(e\left(\frac{1}{8}\right)\) |
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sage:chi.jacobi_sum(n)
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sage:chi.gauss_sum(a)
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pari:znchargauss(g,chi,a)
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sage:chi.jacobi_sum(n)
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sage:chi.kloosterman_sum(a,b)
