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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([17,21]))
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pari:[g,chi] = znchar(Mod(7425,7742))
\(\chi_{7742}(157,\cdot)\)
\(\chi_{7742}(789,\cdot)\)
\(\chi_{7742}(1263,\cdot)\)
\(\chi_{7742}(1895,\cdot)\)
\(\chi_{7742}(2369,\cdot)\)
\(\chi_{7742}(3001,\cdot)\)
\(\chi_{7742}(3475,\cdot)\)
\(\chi_{7742}(4107,\cdot)\)
\(\chi_{7742}(4581,\cdot)\)
\(\chi_{7742}(5687,\cdot)\)
\(\chi_{7742}(6319,\cdot)\)
\(\chi_{7742}(7425,\cdot)\)
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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 ]
\((2845,4901)\) → \((e\left(\frac{17}{42}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 7742 }(7425, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) |
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sage:chi.jacobi_sum(n)
