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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3648, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,27,0,4]))
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pari:[g,chi] = znchar(Mod(175,3648))
\(\chi_{3648}(175,\cdot)\)
\(\chi_{3648}(271,\cdot)\)
\(\chi_{3648}(367,\cdot)\)
\(\chi_{3648}(655,\cdot)\)
\(\chi_{3648}(1327,\cdot)\)
\(\chi_{3648}(1423,\cdot)\)
\(\chi_{3648}(1999,\cdot)\)
\(\chi_{3648}(2095,\cdot)\)
\(\chi_{3648}(2191,\cdot)\)
\(\chi_{3648}(2479,\cdot)\)
\(\chi_{3648}(3151,\cdot)\)
\(\chi_{3648}(3247,\cdot)\)
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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 ]
\((2623,2053,1217,1921)\) → \((-1,-i,1,e\left(\frac{1}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 3648 }(175, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{36}\right)\) |
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sage:chi.jacobi_sum(n)
