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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4056, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([78,78,0,37]))
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pari:[g,chi] = znchar(Mod(67,4056))
\(\chi_{4056}(67,\cdot)\)
\(\chi_{4056}(115,\cdot)\)
\(\chi_{4056}(163,\cdot)\)
\(\chi_{4056}(331,\cdot)\)
\(\chi_{4056}(379,\cdot)\)
\(\chi_{4056}(475,\cdot)\)
\(\chi_{4056}(643,\cdot)\)
\(\chi_{4056}(691,\cdot)\)
\(\chi_{4056}(739,\cdot)\)
\(\chi_{4056}(787,\cdot)\)
\(\chi_{4056}(955,\cdot)\)
\(\chi_{4056}(1003,\cdot)\)
\(\chi_{4056}(1051,\cdot)\)
\(\chi_{4056}(1099,\cdot)\)
\(\chi_{4056}(1267,\cdot)\)
\(\chi_{4056}(1315,\cdot)\)
\(\chi_{4056}(1363,\cdot)\)
\(\chi_{4056}(1411,\cdot)\)
\(\chi_{4056}(1579,\cdot)\)
\(\chi_{4056}(1627,\cdot)\)
\(\chi_{4056}(1675,\cdot)\)
\(\chi_{4056}(1723,\cdot)\)
\(\chi_{4056}(1891,\cdot)\)
\(\chi_{4056}(1987,\cdot)\)
\(\chi_{4056}(2035,\cdot)\)
\(\chi_{4056}(2203,\cdot)\)
\(\chi_{4056}(2251,\cdot)\)
\(\chi_{4056}(2299,\cdot)\)
\(\chi_{4056}(2515,\cdot)\)
\(\chi_{4056}(2563,\cdot)\)
...
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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 ]
\((1015,2029,2705,3889)\) → \((-1,-1,1,e\left(\frac{37}{156}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 4056 }(67, a) \) |
\(1\) | \(1\) | \(e\left(\frac{33}{52}\right)\) | \(e\left(\frac{137}{156}\right)\) | \(e\left(\frac{67}{156}\right)\) | \(e\left(\frac{49}{78}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{25}{52}\right)\) | \(e\left(\frac{20}{39}\right)\) |
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sage:chi.jacobi_sum(n)
