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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3240, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([0,27,17,0]))
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pari:[g,chi] = znchar(Mod(581,3240))
\(\chi_{3240}(101,\cdot)\)
\(\chi_{3240}(221,\cdot)\)
\(\chi_{3240}(461,\cdot)\)
\(\chi_{3240}(581,\cdot)\)
\(\chi_{3240}(821,\cdot)\)
\(\chi_{3240}(941,\cdot)\)
\(\chi_{3240}(1181,\cdot)\)
\(\chi_{3240}(1301,\cdot)\)
\(\chi_{3240}(1541,\cdot)\)
\(\chi_{3240}(1661,\cdot)\)
\(\chi_{3240}(1901,\cdot)\)
\(\chi_{3240}(2021,\cdot)\)
\(\chi_{3240}(2261,\cdot)\)
\(\chi_{3240}(2381,\cdot)\)
\(\chi_{3240}(2621,\cdot)\)
\(\chi_{3240}(2741,\cdot)\)
\(\chi_{3240}(2981,\cdot)\)
\(\chi_{3240}(3101,\cdot)\)
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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 ]
\((2431,1621,3161,1297)\) → \((1,-1,e\left(\frac{17}{54}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3240 }(581, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{37}{54}\right)\) |
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sage:chi.jacobi_sum(n)
