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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4225, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([39,107]))
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pari:[g,chi] = znchar(Mod(7,4225))
\(\chi_{4225}(7,\cdot)\)
\(\chi_{4225}(93,\cdot)\)
\(\chi_{4225}(232,\cdot)\)
\(\chi_{4225}(318,\cdot)\)
\(\chi_{4225}(332,\cdot)\)
\(\chi_{4225}(557,\cdot)\)
\(\chi_{4225}(643,\cdot)\)
\(\chi_{4225}(743,\cdot)\)
\(\chi_{4225}(882,\cdot)\)
\(\chi_{4225}(968,\cdot)\)
\(\chi_{4225}(982,\cdot)\)
\(\chi_{4225}(1068,\cdot)\)
\(\chi_{4225}(1207,\cdot)\)
\(\chi_{4225}(1293,\cdot)\)
\(\chi_{4225}(1307,\cdot)\)
\(\chi_{4225}(1393,\cdot)\)
\(\chi_{4225}(1532,\cdot)\)
\(\chi_{4225}(1618,\cdot)\)
\(\chi_{4225}(1632,\cdot)\)
\(\chi_{4225}(1718,\cdot)\)
\(\chi_{4225}(1857,\cdot)\)
\(\chi_{4225}(1943,\cdot)\)
\(\chi_{4225}(1957,\cdot)\)
\(\chi_{4225}(2043,\cdot)\)
\(\chi_{4225}(2182,\cdot)\)
\(\chi_{4225}(2268,\cdot)\)
\(\chi_{4225}(2282,\cdot)\)
\(\chi_{4225}(2368,\cdot)\)
\(\chi_{4225}(2507,\cdot)\)
\(\chi_{4225}(2593,\cdot)\)
...
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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 ]
\((677,3551)\) → \((i,e\left(\frac{107}{156}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 4225 }(7, a) \) |
\(1\) | \(1\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{125}{156}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{115}{156}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{101}{156}\right)\) | \(e\left(\frac{35}{52}\right)\) | \(e\left(\frac{15}{26}\right)\) |
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sage:chi.jacobi_sum(n)
