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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(15800, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([0,39,0,40]))
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pari:[g,chi] = znchar(Mod(13901,15800))
\(\chi_{15800}(901,\cdot)\)
\(\chi_{15800}(1701,\cdot)\)
\(\chi_{15800}(1901,\cdot)\)
\(\chi_{15800}(3101,\cdot)\)
\(\chi_{15800}(3501,\cdot)\)
\(\chi_{15800}(4101,\cdot)\)
\(\chi_{15800}(4701,\cdot)\)
\(\chi_{15800}(5101,\cdot)\)
\(\chi_{15800}(5501,\cdot)\)
\(\chi_{15800}(5701,\cdot)\)
\(\chi_{15800}(8701,\cdot)\)
\(\chi_{15800}(9101,\cdot)\)
\(\chi_{15800}(9901,\cdot)\)
\(\chi_{15800}(10301,\cdot)\)
\(\chi_{15800}(10501,\cdot)\)
\(\chi_{15800}(10701,\cdot)\)
\(\chi_{15800}(11301,\cdot)\)
\(\chi_{15800}(11701,\cdot)\)
\(\chi_{15800}(11901,\cdot)\)
\(\chi_{15800}(12501,\cdot)\)
\(\chi_{15800}(13901,\cdot)\)
\(\chi_{15800}(14301,\cdot)\)
\(\chi_{15800}(14501,\cdot)\)
\(\chi_{15800}(14901,\cdot)\)
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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 ]
\((3951,7901,11377,12801)\) → \((1,-1,1,e\left(\frac{20}{39}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 15800 }(13901, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{26}\right)\) |
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sage:chi.jacobi_sum(n)
