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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1450, base_ring=CyclotomicField(70))
M = H._module
chi = DirichletCharacter(H, M([49,5]))
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pari:[g,chi] = znchar(Mod(1309,1450))
\(\chi_{1450}(9,\cdot)\)
\(\chi_{1450}(109,\cdot)\)
\(\chi_{1450}(129,\cdot)\)
\(\chi_{1450}(179,\cdot)\)
\(\chi_{1450}(209,\cdot)\)
\(\chi_{1450}(419,\cdot)\)
\(\chi_{1450}(439,\cdot)\)
\(\chi_{1450}(469,\cdot)\)
\(\chi_{1450}(589,\cdot)\)
\(\chi_{1450}(689,\cdot)\)
\(\chi_{1450}(709,\cdot)\)
\(\chi_{1450}(729,\cdot)\)
\(\chi_{1450}(759,\cdot)\)
\(\chi_{1450}(789,\cdot)\)
\(\chi_{1450}(879,\cdot)\)
\(\chi_{1450}(979,\cdot)\)
\(\chi_{1450}(1019,\cdot)\)
\(\chi_{1450}(1079,\cdot)\)
\(\chi_{1450}(1169,\cdot)\)
\(\chi_{1450}(1269,\cdot)\)
\(\chi_{1450}(1289,\cdot)\)
\(\chi_{1450}(1309,\cdot)\)
\(\chi_{1450}(1339,\cdot)\)
\(\chi_{1450}(1369,\cdot)\)
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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 ]
\((1277,901)\) → \((e\left(\frac{7}{10}\right),e\left(\frac{1}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 1450 }(1309, a) \) |
\(1\) | \(1\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{69}{70}\right)\) | \(e\left(\frac{41}{70}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{17}{70}\right)\) | \(e\left(\frac{43}{70}\right)\) | \(e\left(\frac{9}{70}\right)\) | \(e\left(\frac{27}{35}\right)\) |
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sage:chi.jacobi_sum(n)
