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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7056, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,7,0,10]))
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pari:[g,chi] = znchar(Mod(4843,7056))
\(\chi_{7056}(307,\cdot)\)
\(\chi_{7056}(811,\cdot)\)
\(\chi_{7056}(1315,\cdot)\)
\(\chi_{7056}(1819,\cdot)\)
\(\chi_{7056}(2323,\cdot)\)
\(\chi_{7056}(2827,\cdot)\)
\(\chi_{7056}(3835,\cdot)\)
\(\chi_{7056}(4339,\cdot)\)
\(\chi_{7056}(4843,\cdot)\)
\(\chi_{7056}(5347,\cdot)\)
\(\chi_{7056}(5851,\cdot)\)
\(\chi_{7056}(6355,\cdot)\)
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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 ]
\((6175,1765,785,4609)\) → \((-1,i,1,e\left(\frac{5}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 7056 }(4843, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(-i\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(1\) | \(e\left(\frac{19}{28}\right)\) |
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sage:chi.jacobi_sum(n)
