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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3724, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([0,0,1]))
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pari:[g,chi] = znchar(Mod(3137,3724))
\(\chi_{3724}(393,\cdot)\)
\(\chi_{3724}(2157,\cdot)\)
\(\chi_{3724}(2549,\cdot)\)
\(\chi_{3724}(2941,\cdot)\)
\(\chi_{3724}(3137,\cdot)\)
\(\chi_{3724}(3529,\cdot)\)
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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 ]
\((1863,3041,3137)\) → \((1,1,e\left(\frac{1}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 3724 }(3137, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) |
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sage:chi.jacobi_sum(n)
