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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([27,0,41,27]))
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pari:[g,chi] = znchar(Mod(1679,3240))
\(\chi_{3240}(119,\cdot)\)
\(\chi_{3240}(239,\cdot)\)
\(\chi_{3240}(479,\cdot)\)
\(\chi_{3240}(599,\cdot)\)
\(\chi_{3240}(839,\cdot)\)
\(\chi_{3240}(959,\cdot)\)
\(\chi_{3240}(1199,\cdot)\)
\(\chi_{3240}(1319,\cdot)\)
\(\chi_{3240}(1559,\cdot)\)
\(\chi_{3240}(1679,\cdot)\)
\(\chi_{3240}(1919,\cdot)\)
\(\chi_{3240}(2039,\cdot)\)
\(\chi_{3240}(2279,\cdot)\)
\(\chi_{3240}(2399,\cdot)\)
\(\chi_{3240}(2639,\cdot)\)
\(\chi_{3240}(2759,\cdot)\)
\(\chi_{3240}(2999,\cdot)\)
\(\chi_{3240}(3119,\cdot)\)
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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 ]
\((2431,1621,3161,1297)\) → \((-1,1,e\left(\frac{41}{54}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3240 }(1679, a) \) |
\(1\) | \(1\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{13}{54}\right)\) |
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sage:chi.jacobi_sum(n)
