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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([0,39,39,28]))
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pari:[g,chi] = znchar(Mod(7949,15800))
\(\chi_{15800}(1749,\cdot)\)
\(\chi_{15800}(2149,\cdot)\)
\(\chi_{15800}(2949,\cdot)\)
\(\chi_{15800}(3349,\cdot)\)
\(\chi_{15800}(3549,\cdot)\)
\(\chi_{15800}(3749,\cdot)\)
\(\chi_{15800}(4349,\cdot)\)
\(\chi_{15800}(4749,\cdot)\)
\(\chi_{15800}(4949,\cdot)\)
\(\chi_{15800}(5549,\cdot)\)
\(\chi_{15800}(6949,\cdot)\)
\(\chi_{15800}(7349,\cdot)\)
\(\chi_{15800}(7549,\cdot)\)
\(\chi_{15800}(7949,\cdot)\)
\(\chi_{15800}(9749,\cdot)\)
\(\chi_{15800}(10549,\cdot)\)
\(\chi_{15800}(10749,\cdot)\)
\(\chi_{15800}(11949,\cdot)\)
\(\chi_{15800}(12349,\cdot)\)
\(\chi_{15800}(12949,\cdot)\)
\(\chi_{15800}(13549,\cdot)\)
\(\chi_{15800}(13949,\cdot)\)
\(\chi_{15800}(14349,\cdot)\)
\(\chi_{15800}(14549,\cdot)\)
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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 ]
\((3951,7901,11377,12801)\) → \((1,-1,-1,e\left(\frac{14}{39}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 15800 }(7949, a) \) |
\(1\) | \(1\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{41}{78}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{13}\right)\) |
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sage:chi.jacobi_sum(n)
