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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([0,0,1,0]))
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pari:[g,chi] = znchar(Mod(3161,3240))
\(\chi_{3240}(41,\cdot)\)
\(\chi_{3240}(281,\cdot)\)
\(\chi_{3240}(401,\cdot)\)
\(\chi_{3240}(641,\cdot)\)
\(\chi_{3240}(761,\cdot)\)
\(\chi_{3240}(1001,\cdot)\)
\(\chi_{3240}(1121,\cdot)\)
\(\chi_{3240}(1361,\cdot)\)
\(\chi_{3240}(1481,\cdot)\)
\(\chi_{3240}(1721,\cdot)\)
\(\chi_{3240}(1841,\cdot)\)
\(\chi_{3240}(2081,\cdot)\)
\(\chi_{3240}(2201,\cdot)\)
\(\chi_{3240}(2441,\cdot)\)
\(\chi_{3240}(2561,\cdot)\)
\(\chi_{3240}(2801,\cdot)\)
\(\chi_{3240}(2921,\cdot)\)
\(\chi_{3240}(3161,\cdot)\)
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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 ]
\((2431,1621,3161,1297)\) → \((1,1,e\left(\frac{1}{54}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3240 }(3161, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{53}{54}\right)\) |
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sage:chi.jacobi_sum(n)
