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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3332, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,8,27]))
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pari:[g,chi] = znchar(Mod(31,3332))
\(\chi_{3332}(31,\cdot)\)
\(\chi_{3332}(215,\cdot)\)
\(\chi_{3332}(227,\cdot)\)
\(\chi_{3332}(411,\cdot)\)
\(\chi_{3332}(607,\cdot)\)
\(\chi_{3332}(619,\cdot)\)
\(\chi_{3332}(1195,\cdot)\)
\(\chi_{3332}(1391,\cdot)\)
\(\chi_{3332}(1587,\cdot)\)
\(\chi_{3332}(1795,\cdot)\)
\(\chi_{3332}(1979,\cdot)\)
\(\chi_{3332}(2187,\cdot)\)
\(\chi_{3332}(2383,\cdot)\)
\(\chi_{3332}(2579,\cdot)\)
\(\chi_{3332}(3155,\cdot)\)
\(\chi_{3332}(3167,\cdot)\)
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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 ]
\((1667,885,785)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{9}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 3332 }(31, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(-i\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{11}{16}\right)\) |
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sage:chi.jacobi_sum(n)
