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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7728, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,0,0,11,12]))
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pari:[g,chi] = znchar(Mod(3583,7728))
\(\chi_{7728}(223,\cdot)\)
\(\chi_{7728}(1231,\cdot)\)
\(\chi_{7728}(1567,\cdot)\)
\(\chi_{7728}(2239,\cdot)\)
\(\chi_{7728}(2911,\cdot)\)
\(\chi_{7728}(3247,\cdot)\)
\(\chi_{7728}(3583,\cdot)\)
\(\chi_{7728}(3919,\cdot)\)
\(\chi_{7728}(6607,\cdot)\)
\(\chi_{7728}(7615,\cdot)\)
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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 ]
\((4831,5797,5153,6625,6721)\) → \((-1,1,1,-1,e\left(\frac{6}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 7728 }(3583, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{1}{22}\right)\) |
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sage:chi.jacobi_sum(n)
