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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([27,0,28,27]))
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pari:[g,chi] = znchar(Mod(79,3240))
\(\chi_{3240}(79,\cdot)\)
\(\chi_{3240}(319,\cdot)\)
\(\chi_{3240}(439,\cdot)\)
\(\chi_{3240}(679,\cdot)\)
\(\chi_{3240}(799,\cdot)\)
\(\chi_{3240}(1039,\cdot)\)
\(\chi_{3240}(1159,\cdot)\)
\(\chi_{3240}(1399,\cdot)\)
\(\chi_{3240}(1519,\cdot)\)
\(\chi_{3240}(1759,\cdot)\)
\(\chi_{3240}(1879,\cdot)\)
\(\chi_{3240}(2119,\cdot)\)
\(\chi_{3240}(2239,\cdot)\)
\(\chi_{3240}(2479,\cdot)\)
\(\chi_{3240}(2599,\cdot)\)
\(\chi_{3240}(2839,\cdot)\)
\(\chi_{3240}(2959,\cdot)\)
\(\chi_{3240}(3199,\cdot)\)
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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 ]
\((2431,1621,3161,1297)\) → \((-1,1,e\left(\frac{14}{27}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3240 }(79, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{13}{27}\right)\) |
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sage:chi.jacobi_sum(n)
