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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1045, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,6,10]))
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pari:[g,chi] = znchar(Mod(26,1045))
\(\chi_{1045}(26,\cdot)\)
\(\chi_{1045}(201,\cdot)\)
\(\chi_{1045}(311,\cdot)\)
\(\chi_{1045}(581,\cdot)\)
\(\chi_{1045}(676,\cdot)\)
\(\chi_{1045}(691,\cdot)\)
\(\chi_{1045}(786,\cdot)\)
\(\chi_{1045}(961,\cdot)\)
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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 ]
\((837,761,496)\) → \((1,e\left(\frac{1}{5}\right),e\left(\frac{1}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 1045 }(26, a) \) |
\(1\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) |
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sage:chi.jacobi_sum(n)
