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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3888, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([27,27,1]))
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pari:[g,chi] = znchar(Mod(1223,3888))
\(\chi_{3888}(71,\cdot)\)
\(\chi_{3888}(359,\cdot)\)
\(\chi_{3888}(503,\cdot)\)
\(\chi_{3888}(791,\cdot)\)
\(\chi_{3888}(935,\cdot)\)
\(\chi_{3888}(1223,\cdot)\)
\(\chi_{3888}(1367,\cdot)\)
\(\chi_{3888}(1655,\cdot)\)
\(\chi_{3888}(1799,\cdot)\)
\(\chi_{3888}(2087,\cdot)\)
\(\chi_{3888}(2231,\cdot)\)
\(\chi_{3888}(2519,\cdot)\)
\(\chi_{3888}(2663,\cdot)\)
\(\chi_{3888}(2951,\cdot)\)
\(\chi_{3888}(3095,\cdot)\)
\(\chi_{3888}(3383,\cdot)\)
\(\chi_{3888}(3527,\cdot)\)
\(\chi_{3888}(3815,\cdot)\)
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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 ]
\((2431,2917,1217)\) → \((-1,-1,e\left(\frac{1}{54}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 3888 }(1223, a) \) |
\(1\) | \(1\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{47}{54}\right)\) |
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sage:chi.jacobi_sum(n)
