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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2032, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([63,63,55]))
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pari:[g,chi] = znchar(Mod(1335,2032))
\(\chi_{2032}(7,\cdot)\)
\(\chi_{2032}(23,\cdot)\)
\(\chi_{2032}(39,\cdot)\)
\(\chi_{2032}(55,\cdot)\)
\(\chi_{2032}(183,\cdot)\)
\(\chi_{2032}(311,\cdot)\)
\(\chi_{2032}(439,\cdot)\)
\(\chi_{2032}(487,\cdot)\)
\(\chi_{2032}(551,\cdot)\)
\(\chi_{2032}(599,\cdot)\)
\(\chi_{2032}(647,\cdot)\)
\(\chi_{2032}(727,\cdot)\)
\(\chi_{2032}(791,\cdot)\)
\(\chi_{2032}(807,\cdot)\)
\(\chi_{2032}(855,\cdot)\)
\(\chi_{2032}(871,\cdot)\)
\(\chi_{2032}(903,\cdot)\)
\(\chi_{2032}(935,\cdot)\)
\(\chi_{2032}(967,\cdot)\)
\(\chi_{2032}(999,\cdot)\)
\(\chi_{2032}(1191,\cdot)\)
\(\chi_{2032}(1239,\cdot)\)
\(\chi_{2032}(1255,\cdot)\)
\(\chi_{2032}(1335,\cdot)\)
\(\chi_{2032}(1367,\cdot)\)
\(\chi_{2032}(1511,\cdot)\)
\(\chi_{2032}(1527,\cdot)\)
\(\chi_{2032}(1591,\cdot)\)
\(\chi_{2032}(1607,\cdot)\)
\(\chi_{2032}(1767,\cdot)\)
...
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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 ]
\((255,1525,257)\) → \((-1,-1,e\left(\frac{55}{126}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 2032 }(1335, a) \) |
\(1\) | \(1\) | \(e\left(\frac{55}{126}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{67}{126}\right)\) | \(e\left(\frac{115}{126}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{17}{126}\right)\) |
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sage:chi.jacobi_sum(n)
