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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8041, base_ring=CyclotomicField(70))
M = H._module
chi = DirichletCharacter(H, M([49,0,20]))
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pari:[g,chi] = znchar(Mod(4132,8041))
\(\chi_{8041}(35,\cdot)\)
\(\chi_{8041}(613,\cdot)\)
\(\chi_{8041}(766,\cdot)\)
\(\chi_{8041}(919,\cdot)\)
\(\chi_{8041}(987,\cdot)\)
\(\chi_{8041}(1344,\cdot)\)
\(\chi_{8041}(1718,\cdot)\)
\(\chi_{8041}(2075,\cdot)\)
\(\chi_{8041}(2228,\cdot)\)
\(\chi_{8041}(2449,\cdot)\)
\(\chi_{8041}(2670,\cdot)\)
\(\chi_{8041}(3401,\cdot)\)
\(\chi_{8041}(3418,\cdot)\)
\(\chi_{8041}(3537,\cdot)\)
\(\chi_{8041}(3911,\cdot)\)
\(\chi_{8041}(4132,\cdot)\)
\(\chi_{8041}(4149,\cdot)\)
\(\chi_{8041}(4880,\cdot)\)
\(\chi_{8041}(5594,\cdot)\)
\(\chi_{8041}(6036,\cdot)\)
\(\chi_{8041}(6342,\cdot)\)
\(\chi_{8041}(6767,\cdot)\)
\(\chi_{8041}(7345,\cdot)\)
\(\chi_{8041}(7498,\cdot)\)
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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 ]
\((6580,2366,562)\) → \((e\left(\frac{7}{10}\right),1,e\left(\frac{2}{7}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 8041 }(4132, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{29}{70}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{17}{70}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{5}{7}\right)\) |
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sage:chi.jacobi_sum(n)
