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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8112, base_ring=CyclotomicField(52))
M = H._module
chi = DirichletCharacter(H, M([0,13,0,42]))
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pari:[g,chi] = znchar(Mod(181,8112))
\(\chi_{8112}(181,\cdot)\)
\(\chi_{8112}(493,\cdot)\)
\(\chi_{8112}(805,\cdot)\)
\(\chi_{8112}(1117,\cdot)\)
\(\chi_{8112}(1429,\cdot)\)
\(\chi_{8112}(1741,\cdot)\)
\(\chi_{8112}(2053,\cdot)\)
\(\chi_{8112}(2677,\cdot)\)
\(\chi_{8112}(2989,\cdot)\)
\(\chi_{8112}(3301,\cdot)\)
\(\chi_{8112}(3613,\cdot)\)
\(\chi_{8112}(3925,\cdot)\)
\(\chi_{8112}(4237,\cdot)\)
\(\chi_{8112}(4549,\cdot)\)
\(\chi_{8112}(4861,\cdot)\)
\(\chi_{8112}(5173,\cdot)\)
\(\chi_{8112}(5485,\cdot)\)
\(\chi_{8112}(5797,\cdot)\)
\(\chi_{8112}(6109,\cdot)\)
\(\chi_{8112}(6733,\cdot)\)
\(\chi_{8112}(7045,\cdot)\)
\(\chi_{8112}(7357,\cdot)\)
\(\chi_{8112}(7669,\cdot)\)
\(\chi_{8112}(7981,\cdot)\)
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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 ]
\((5071,6085,2705,3889)\) → \((1,i,1,e\left(\frac{21}{26}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 8112 }(181, a) \) |
\(1\) | \(1\) | \(e\left(\frac{27}{52}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{23}{52}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(i\) | \(-1\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{3}{52}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{23}{52}\right)\) |
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sage:chi.jacobi_sum(n)
