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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4056, base_ring=CyclotomicField(52))
M = H._module
chi = DirichletCharacter(H, M([26,0,0,7]))
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pari:[g,chi] = znchar(Mod(31,4056))
\(\chi_{4056}(31,\cdot)\)
\(\chi_{4056}(151,\cdot)\)
\(\chi_{4056}(343,\cdot)\)
\(\chi_{4056}(463,\cdot)\)
\(\chi_{4056}(655,\cdot)\)
\(\chi_{4056}(967,\cdot)\)
\(\chi_{4056}(1087,\cdot)\)
\(\chi_{4056}(1279,\cdot)\)
\(\chi_{4056}(1399,\cdot)\)
\(\chi_{4056}(1711,\cdot)\)
\(\chi_{4056}(1903,\cdot)\)
\(\chi_{4056}(2023,\cdot)\)
\(\chi_{4056}(2215,\cdot)\)
\(\chi_{4056}(2335,\cdot)\)
\(\chi_{4056}(2527,\cdot)\)
\(\chi_{4056}(2647,\cdot)\)
\(\chi_{4056}(2839,\cdot)\)
\(\chi_{4056}(2959,\cdot)\)
\(\chi_{4056}(3151,\cdot)\)
\(\chi_{4056}(3271,\cdot)\)
\(\chi_{4056}(3463,\cdot)\)
\(\chi_{4056}(3583,\cdot)\)
\(\chi_{4056}(3775,\cdot)\)
\(\chi_{4056}(3895,\cdot)\)
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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 ]
\((1015,2029,2705,3889)\) → \((-1,1,1,e\left(\frac{7}{52}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 4056 }(31, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{52}\right)\) | \(e\left(\frac{47}{52}\right)\) | \(e\left(\frac{19}{52}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(i\) | \(1\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{17}{52}\right)\) | \(e\left(\frac{3}{26}\right)\) |
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sage:chi.jacobi_sum(n)
