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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(33957, base_ring=CyclotomicField(1470))
M = H._module
chi = DirichletCharacter(H, M([735,620,882]))
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pari:[g,chi] = znchar(Mod(746,33957))
\(\chi_{33957}(53,\cdot)\)
\(\chi_{33957}(170,\cdot)\)
\(\chi_{33957}(179,\cdot)\)
\(\chi_{33957}(368,\cdot)\)
\(\chi_{33957}(548,\cdot)\)
\(\chi_{33957}(620,\cdot)\)
\(\chi_{33957}(674,\cdot)\)
\(\chi_{33957}(746,\cdot)\)
\(\chi_{33957}(872,\cdot)\)
\(\chi_{33957}(1061,\cdot)\)
\(\chi_{33957}(1115,\cdot)\)
\(\chi_{33957}(1241,\cdot)\)
\(\chi_{33957}(1313,\cdot)\)
\(\chi_{33957}(1367,\cdot)\)
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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 ]
\((18866,14752,24697)\) → \((-1,e\left(\frac{62}{147}\right),e\left(\frac{3}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
| \( \chi_{ 33957 }(746, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1357}{1470}\right)\) | \(e\left(\frac{622}{735}\right)\) | \(e\left(\frac{193}{1470}\right)\) | \(e\left(\frac{377}{490}\right)\) | \(e\left(\frac{8}{147}\right)\) | \(e\left(\frac{197}{245}\right)\) | \(e\left(\frac{509}{735}\right)\) | \(e\left(\frac{653}{1470}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{479}{490}\right)\) |
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sage:chi.jacobi_sum(n)
