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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([21,0,21,21,38]))
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pari:[g,chi] = znchar(Mod(4919,5160))
\(\chi_{5160}(239,\cdot)\)
\(\chi_{5160}(359,\cdot)\)
\(\chi_{5160}(599,\cdot)\)
\(\chi_{5160}(719,\cdot)\)
\(\chi_{5160}(959,\cdot)\)
\(\chi_{5160}(1199,\cdot)\)
\(\chi_{5160}(2159,\cdot)\)
\(\chi_{5160}(2519,\cdot)\)
\(\chi_{5160}(3119,\cdot)\)
\(\chi_{5160}(3239,\cdot)\)
\(\chi_{5160}(4439,\cdot)\)
\(\chi_{5160}(4919,\cdot)\)
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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 ]
\((3871,2581,1721,3097,4561)\) → \((-1,1,-1,-1,e\left(\frac{19}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 5160 }(4919, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{14}\right)\) |
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sage:chi.jacobi_sum(n)
