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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8041, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,0,1]))
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pari:[g,chi] = znchar(Mod(562,8041))
\(\chi_{8041}(562,\cdot)\)
\(\chi_{8041}(749,\cdot)\)
\(\chi_{8041}(936,\cdot)\)
\(\chi_{8041}(1123,\cdot)\)
\(\chi_{8041}(1310,\cdot)\)
\(\chi_{8041}(2993,\cdot)\)
\(\chi_{8041}(3554,\cdot)\)
\(\chi_{8041}(4115,\cdot)\)
\(\chi_{8041}(5050,\cdot)\)
\(\chi_{8041}(5237,\cdot)\)
\(\chi_{8041}(7107,\cdot)\)
\(\chi_{8041}(7855,\cdot)\)
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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 ]
\((6580,2366,562)\) → \((1,1,e\left(\frac{1}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 8041 }(562, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{13}{42}\right)\) |
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sage:chi.jacobi_sum(n)
