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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([0,0,21,21,37]))
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pari:[g,chi] = znchar(Mod(2729,5160))
\(\chi_{5160}(89,\cdot)\)
\(\chi_{5160}(329,\cdot)\)
\(\chi_{5160}(449,\cdot)\)
\(\chi_{5160}(929,\cdot)\)
\(\chi_{5160}(1409,\cdot)\)
\(\chi_{5160}(2609,\cdot)\)
\(\chi_{5160}(2729,\cdot)\)
\(\chi_{5160}(3329,\cdot)\)
\(\chi_{5160}(3689,\cdot)\)
\(\chi_{5160}(4649,\cdot)\)
\(\chi_{5160}(4889,\cdot)\)
\(\chi_{5160}(5129,\cdot)\)
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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 ]
\((3871,2581,1721,3097,4561)\) → \((1,1,-1,-1,e\left(\frac{37}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 5160 }(2729, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{14}\right)\) |
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sage:chi.jacobi_sum(n)
