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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7742, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([40,28]))
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pari:[g,chi] = znchar(Mod(7165,7742))
\(\chi_{7742}(23,\cdot)\)
\(\chi_{7742}(529,\cdot)\)
\(\chi_{7742}(1129,\cdot)\)
\(\chi_{7742}(2741,\cdot)\)
\(\chi_{7742}(3341,\cdot)\)
\(\chi_{7742}(3847,\cdot)\)
\(\chi_{7742}(4447,\cdot)\)
\(\chi_{7742}(4953,\cdot)\)
\(\chi_{7742}(5553,\cdot)\)
\(\chi_{7742}(6059,\cdot)\)
\(\chi_{7742}(6659,\cdot)\)
\(\chi_{7742}(7165,\cdot)\)
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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 ]
\((2845,4901)\) → \((e\left(\frac{20}{21}\right),e\left(\frac{2}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 7742 }(7165, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) |
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sage:chi.jacobi_sum(n)
