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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5160, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([0,14,0,7,2]))
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pari:[g,chi] = znchar(Mod(3037,5160))
\(\chi_{5160}(733,\cdot)\)
\(\chi_{5160}(973,\cdot)\)
\(\chi_{5160}(997,\cdot)\)
\(\chi_{5160}(1957,\cdot)\)
\(\chi_{5160}(2053,\cdot)\)
\(\chi_{5160}(2533,\cdot)\)
\(\chi_{5160}(2797,\cdot)\)
\(\chi_{5160}(3037,\cdot)\)
\(\chi_{5160}(4093,\cdot)\)
\(\chi_{5160}(4117,\cdot)\)
\(\chi_{5160}(4597,\cdot)\)
\(\chi_{5160}(5053,\cdot)\)
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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 ]
\((3871,2581,1721,3097,4561)\) → \((1,-1,1,i,e\left(\frac{1}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 5160 }(3037, a) \) |
\(1\) | \(1\) | \(-i\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(i\) | \(e\left(\frac{3}{7}\right)\) |
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sage:chi.jacobi_sum(n)
