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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([1,0]))
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pari:[g,chi] = znchar(Mod(32,3751))
\(\chi_{3751}(32,\cdot)\)
\(\chi_{3751}(373,\cdot)\)
\(\chi_{3751}(714,\cdot)\)
\(\chi_{3751}(1055,\cdot)\)
\(\chi_{3751}(1396,\cdot)\)
\(\chi_{3751}(1737,\cdot)\)
\(\chi_{3751}(2078,\cdot)\)
\(\chi_{3751}(2760,\cdot)\)
\(\chi_{3751}(3101,\cdot)\)
\(\chi_{3751}(3442,\cdot)\)
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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 ]
\((2543,2421)\) → \((e\left(\frac{1}{22}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 3751 }(32, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{22}\right)\) | \(1\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(1\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) |
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sage:chi.jacobi_sum(n)
