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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3648, base_ring=CyclotomicField(144))
M = H._module
chi = DirichletCharacter(H, M([72,135,0,16]))
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pari:[g,chi] = znchar(Mod(2227,3648))
\(\chi_{3648}(43,\cdot)\)
\(\chi_{3648}(139,\cdot)\)
\(\chi_{3648}(187,\cdot)\)
\(\chi_{3648}(283,\cdot)\)
\(\chi_{3648}(403,\cdot)\)
\(\chi_{3648}(427,\cdot)\)
\(\chi_{3648}(499,\cdot)\)
\(\chi_{3648}(595,\cdot)\)
\(\chi_{3648}(643,\cdot)\)
\(\chi_{3648}(739,\cdot)\)
\(\chi_{3648}(859,\cdot)\)
\(\chi_{3648}(883,\cdot)\)
\(\chi_{3648}(955,\cdot)\)
\(\chi_{3648}(1051,\cdot)\)
\(\chi_{3648}(1099,\cdot)\)
\(\chi_{3648}(1195,\cdot)\)
\(\chi_{3648}(1315,\cdot)\)
\(\chi_{3648}(1339,\cdot)\)
\(\chi_{3648}(1411,\cdot)\)
\(\chi_{3648}(1507,\cdot)\)
\(\chi_{3648}(1555,\cdot)\)
\(\chi_{3648}(1651,\cdot)\)
\(\chi_{3648}(1771,\cdot)\)
\(\chi_{3648}(1795,\cdot)\)
\(\chi_{3648}(1867,\cdot)\)
\(\chi_{3648}(1963,\cdot)\)
\(\chi_{3648}(2011,\cdot)\)
\(\chi_{3648}(2107,\cdot)\)
\(\chi_{3648}(2227,\cdot)\)
\(\chi_{3648}(2251,\cdot)\)
...
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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 ]
\((2623,2053,1217,1921)\) → \((-1,e\left(\frac{15}{16}\right),1,e\left(\frac{1}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 3648 }(2227, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{103}{144}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{89}{144}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{61}{72}\right)\) | \(e\left(\frac{31}{72}\right)\) | \(e\left(\frac{29}{144}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{37}{144}\right)\) |
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sage:chi.jacobi_sum(n)
