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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([63,0,55]))
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pari:[g,chi] = znchar(Mod(319,2032))
\(\chi_{2032}(175,\cdot)\)
\(\chi_{2032}(223,\cdot)\)
\(\chi_{2032}(239,\cdot)\)
\(\chi_{2032}(319,\cdot)\)
\(\chi_{2032}(351,\cdot)\)
\(\chi_{2032}(495,\cdot)\)
\(\chi_{2032}(511,\cdot)\)
\(\chi_{2032}(575,\cdot)\)
\(\chi_{2032}(591,\cdot)\)
\(\chi_{2032}(751,\cdot)\)
\(\chi_{2032}(815,\cdot)\)
\(\chi_{2032}(847,\cdot)\)
\(\chi_{2032}(863,\cdot)\)
\(\chi_{2032}(895,\cdot)\)
\(\chi_{2032}(975,\cdot)\)
\(\chi_{2032}(1007,\cdot)\)
\(\chi_{2032}(1023,\cdot)\)
\(\chi_{2032}(1039,\cdot)\)
\(\chi_{2032}(1055,\cdot)\)
\(\chi_{2032}(1071,\cdot)\)
\(\chi_{2032}(1199,\cdot)\)
\(\chi_{2032}(1327,\cdot)\)
\(\chi_{2032}(1455,\cdot)\)
\(\chi_{2032}(1503,\cdot)\)
\(\chi_{2032}(1567,\cdot)\)
\(\chi_{2032}(1615,\cdot)\)
\(\chi_{2032}(1663,\cdot)\)
\(\chi_{2032}(1743,\cdot)\)
\(\chi_{2032}(1807,\cdot)\)
\(\chi_{2032}(1823,\cdot)\)
...
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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 ]
\((255,1525,257)\) → \((-1,1,e\left(\frac{55}{126}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 2032 }(319, a) \) |
\(1\) | \(1\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{23}{126}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{115}{126}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{40}{63}\right)\) |
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sage:chi.jacobi_sum(n)
