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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8112, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([78,117,0,85]))
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pari:[g,chi] = znchar(Mod(4435,8112))
\(\chi_{8112}(67,\cdot)\)
\(\chi_{8112}(475,\cdot)\)
\(\chi_{8112}(643,\cdot)\)
\(\chi_{8112}(691,\cdot)\)
\(\chi_{8112}(1051,\cdot)\)
\(\chi_{8112}(1099,\cdot)\)
\(\chi_{8112}(1267,\cdot)\)
\(\chi_{8112}(1315,\cdot)\)
\(\chi_{8112}(1675,\cdot)\)
\(\chi_{8112}(1723,\cdot)\)
\(\chi_{8112}(1891,\cdot)\)
\(\chi_{8112}(2299,\cdot)\)
\(\chi_{8112}(2515,\cdot)\)
\(\chi_{8112}(2563,\cdot)\)
\(\chi_{8112}(2923,\cdot)\)
\(\chi_{8112}(2971,\cdot)\)
\(\chi_{8112}(3139,\cdot)\)
\(\chi_{8112}(3187,\cdot)\)
\(\chi_{8112}(3547,\cdot)\)
\(\chi_{8112}(3595,\cdot)\)
\(\chi_{8112}(3763,\cdot)\)
\(\chi_{8112}(3811,\cdot)\)
\(\chi_{8112}(4171,\cdot)\)
\(\chi_{8112}(4219,\cdot)\)
\(\chi_{8112}(4387,\cdot)\)
\(\chi_{8112}(4435,\cdot)\)
\(\chi_{8112}(4795,\cdot)\)
\(\chi_{8112}(4843,\cdot)\)
\(\chi_{8112}(5011,\cdot)\)
\(\chi_{8112}(5059,\cdot)\)
...
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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 ]
\((5071,6085,2705,3889)\) → \((-1,-i,1,e\left(\frac{85}{156}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 8112 }(4435, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{47}{156}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{7}{156}\right)\) | \(e\left(\frac{49}{52}\right)\) | \(e\left(\frac{149}{156}\right)\) |
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sage:chi.jacobi_sum(n)
