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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1323, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([28,18]))
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pari:[g,chi] = znchar(Mod(127,1323))
\(\chi_{1323}(64,\cdot)\)
\(\chi_{1323}(127,\cdot)\)
\(\chi_{1323}(253,\cdot)\)
\(\chi_{1323}(316,\cdot)\)
\(\chi_{1323}(505,\cdot)\)
\(\chi_{1323}(631,\cdot)\)
\(\chi_{1323}(694,\cdot)\)
\(\chi_{1323}(820,\cdot)\)
\(\chi_{1323}(1009,\cdot)\)
\(\chi_{1323}(1072,\cdot)\)
\(\chi_{1323}(1198,\cdot)\)
\(\chi_{1323}(1261,\cdot)\)
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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 ]
\((785,1081)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{3}{7}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1323 }(127, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(1\) |
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sage:chi.jacobi_sum(n)
