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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([0,1]))
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pari:[g,chi] = znchar(Mod(1081,1323))
\(\chi_{1323}(82,\cdot)\)
\(\chi_{1323}(136,\cdot)\)
\(\chi_{1323}(271,\cdot)\)
\(\chi_{1323}(514,\cdot)\)
\(\chi_{1323}(649,\cdot)\)
\(\chi_{1323}(703,\cdot)\)
\(\chi_{1323}(838,\cdot)\)
\(\chi_{1323}(892,\cdot)\)
\(\chi_{1323}(1027,\cdot)\)
\(\chi_{1323}(1081,\cdot)\)
\(\chi_{1323}(1216,\cdot)\)
\(\chi_{1323}(1270,\cdot)\)
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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 ]
\((785,1081)\) → \((1,e\left(\frac{1}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1323 }(1081, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{5}{6}\right)\) |
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sage:chi.jacobi_sum(n)
