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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2394, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([15,9,13]))
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pari:[g,chi] = znchar(Mod(41,2394))
\(\chi_{2394}(41,\cdot)\)
\(\chi_{2394}(713,\cdot)\)
\(\chi_{2394}(965,\cdot)\)
\(\chi_{2394}(1343,\cdot)\)
\(\chi_{2394}(1427,\cdot)\)
\(\chi_{2394}(1553,\cdot)\)
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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 ]
\((533,1711,1009)\) → \((e\left(\frac{5}{6}\right),-1,e\left(\frac{13}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 2394 }(41, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{2}{9}\right)\) | \(-1\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(1\) | \(-1\) | \(e\left(\frac{1}{18}\right)\) |
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sage:chi.jacobi_sum(n)
