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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1710, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([6,9,16]))
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pari:[g,chi] = znchar(Mod(47,1710))
\(\chi_{1710}(47,\cdot)\)
\(\chi_{1710}(137,\cdot)\)
\(\chi_{1710}(347,\cdot)\)
\(\chi_{1710}(443,\cdot)\)
\(\chi_{1710}(473,\cdot)\)
\(\chi_{1710}(833,\cdot)\)
\(\chi_{1710}(1073,\cdot)\)
\(\chi_{1710}(1127,\cdot)\)
\(\chi_{1710}(1157,\cdot)\)
\(\chi_{1710}(1163,\cdot)\)
\(\chi_{1710}(1373,\cdot)\)
\(\chi_{1710}(1517,\cdot)\)
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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 ]
\((191,1027,1351)\) → \((e\left(\frac{1}{6}\right),i,e\left(\frac{4}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 1710 }(47, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(-1\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(1\) | \(i\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{19}{36}\right)\) |
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sage:chi.jacobi_sum(n)
