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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5616, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([0,0,16,9]))
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pari:[g,chi] = znchar(Mod(4705,5616))
\(\chi_{5616}(337,\cdot)\)
\(\chi_{5616}(961,\cdot)\)
\(\chi_{5616}(2209,\cdot)\)
\(\chi_{5616}(2833,\cdot)\)
\(\chi_{5616}(4081,\cdot)\)
\(\chi_{5616}(4705,\cdot)\)
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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 ]
\((703,4213,2081,3889)\) → \((1,1,e\left(\frac{8}{9}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 5616 }(4705, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) |
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sage:chi.jacobi_sum(n)
