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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3751, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([20,0]))
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pari:[g,chi] = znchar(Mod(2729,3751))
\(\chi_{3751}(342,\cdot)\)
\(\chi_{3751}(683,\cdot)\)
\(\chi_{3751}(1024,\cdot)\)
\(\chi_{3751}(1365,\cdot)\)
\(\chi_{3751}(1706,\cdot)\)
\(\chi_{3751}(2047,\cdot)\)
\(\chi_{3751}(2388,\cdot)\)
\(\chi_{3751}(2729,\cdot)\)
\(\chi_{3751}(3070,\cdot)\)
\(\chi_{3751}(3411,\cdot)\)
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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 ]
\((2543,2421)\) → \((e\left(\frac{10}{11}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 3751 }(2729, a) \) |
\(1\) | \(1\) | \(e\left(\frac{10}{11}\right)\) | \(1\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(1\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) |
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sage:chi.jacobi_sum(n)
