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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3648, base_ring=CyclotomicField(72))
M = H._module
chi = DirichletCharacter(H, M([36,45,0,20]))
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pari:[g,chi] = znchar(Mod(583,3648))
\(\chi_{3648}(295,\cdot)\)
\(\chi_{3648}(439,\cdot)\)
\(\chi_{3648}(535,\cdot)\)
\(\chi_{3648}(583,\cdot)\)
\(\chi_{3648}(679,\cdot)\)
\(\chi_{3648}(775,\cdot)\)
\(\chi_{3648}(1207,\cdot)\)
\(\chi_{3648}(1351,\cdot)\)
\(\chi_{3648}(1447,\cdot)\)
\(\chi_{3648}(1495,\cdot)\)
\(\chi_{3648}(1591,\cdot)\)
\(\chi_{3648}(1687,\cdot)\)
\(\chi_{3648}(2119,\cdot)\)
\(\chi_{3648}(2263,\cdot)\)
\(\chi_{3648}(2359,\cdot)\)
\(\chi_{3648}(2407,\cdot)\)
\(\chi_{3648}(2503,\cdot)\)
\(\chi_{3648}(2599,\cdot)\)
\(\chi_{3648}(3031,\cdot)\)
\(\chi_{3648}(3175,\cdot)\)
\(\chi_{3648}(3271,\cdot)\)
\(\chi_{3648}(3319,\cdot)\)
\(\chi_{3648}(3415,\cdot)\)
\(\chi_{3648}(3511,\cdot)\)
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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 ]
\((2623,2053,1217,1921)\) → \((-1,e\left(\frac{5}{8}\right),1,e\left(\frac{5}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 3648 }(583, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{72}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{55}{72}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{43}{72}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{35}{72}\right)\) |
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sage:chi.jacobi_sum(n)
