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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([27,27,31,0]))
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pari:[g,chi] = znchar(Mod(2171,3240))
\(\chi_{3240}(11,\cdot)\)
\(\chi_{3240}(131,\cdot)\)
\(\chi_{3240}(371,\cdot)\)
\(\chi_{3240}(491,\cdot)\)
\(\chi_{3240}(731,\cdot)\)
\(\chi_{3240}(851,\cdot)\)
\(\chi_{3240}(1091,\cdot)\)
\(\chi_{3240}(1211,\cdot)\)
\(\chi_{3240}(1451,\cdot)\)
\(\chi_{3240}(1571,\cdot)\)
\(\chi_{3240}(1811,\cdot)\)
\(\chi_{3240}(1931,\cdot)\)
\(\chi_{3240}(2171,\cdot)\)
\(\chi_{3240}(2291,\cdot)\)
\(\chi_{3240}(2531,\cdot)\)
\(\chi_{3240}(2651,\cdot)\)
\(\chi_{3240}(2891,\cdot)\)
\(\chi_{3240}(3011,\cdot)\)
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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 ]
\((2431,1621,3161,1297)\) → \((-1,-1,e\left(\frac{31}{54}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3240 }(2171, a) \) |
\(1\) | \(1\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{23}{54}\right)\) |
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sage:chi.jacobi_sum(n)
