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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5160, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,0,21,0,20]))
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pari:[g,chi] = znchar(Mod(4271,5160))
\(\chi_{5160}(311,\cdot)\)
\(\chi_{5160}(791,\cdot)\)
\(\chi_{5160}(1271,\cdot)\)
\(\chi_{5160}(1391,\cdot)\)
\(\chi_{5160}(1631,\cdot)\)
\(\chi_{5160}(1751,\cdot)\)
\(\chi_{5160}(1991,\cdot)\)
\(\chi_{5160}(2231,\cdot)\)
\(\chi_{5160}(3191,\cdot)\)
\(\chi_{5160}(3551,\cdot)\)
\(\chi_{5160}(4151,\cdot)\)
\(\chi_{5160}(4271,\cdot)\)
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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 ]
\((3871,2581,1721,3097,4561)\) → \((-1,1,-1,1,e\left(\frac{10}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 5160 }(4271, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{14}\right)\) |
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sage:chi.jacobi_sum(n)
