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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3240, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,18,20,9]))
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pari:[g,chi] = znchar(Mod(397,3240))
\(\chi_{3240}(37,\cdot)\)
\(\chi_{3240}(253,\cdot)\)
\(\chi_{3240}(397,\cdot)\)
\(\chi_{3240}(613,\cdot)\)
\(\chi_{3240}(1117,\cdot)\)
\(\chi_{3240}(1333,\cdot)\)
\(\chi_{3240}(1477,\cdot)\)
\(\chi_{3240}(1693,\cdot)\)
\(\chi_{3240}(2197,\cdot)\)
\(\chi_{3240}(2413,\cdot)\)
\(\chi_{3240}(2557,\cdot)\)
\(\chi_{3240}(2773,\cdot)\)
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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 ]
\((2431,1621,3161,1297)\) → \((1,-1,e\left(\frac{5}{9}\right),i)\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3240 }(397, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{4}{9}\right)\) |
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sage:chi.jacobi_sum(n)
