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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3150, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([5,21,0]))
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pari:[g,chi] = znchar(Mod(659,3150))
\(\chi_{3150}(29,\cdot)\)
\(\chi_{3150}(239,\cdot)\)
\(\chi_{3150}(659,\cdot)\)
\(\chi_{3150}(869,\cdot)\)
\(\chi_{3150}(1289,\cdot)\)
\(\chi_{3150}(1919,\cdot)\)
\(\chi_{3150}(2129,\cdot)\)
\(\chi_{3150}(2759,\cdot)\)
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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 ]
\((2801,127,451)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{7}{10}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 3150 }(659, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{1}{6}\right)\) |
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sage:chi.jacobi_sum(n)
