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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3150, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([10,27,40]))
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pari:[g,chi] = znchar(Mod(137,3150))
\(\chi_{3150}(23,\cdot)\)
\(\chi_{3150}(137,\cdot)\)
\(\chi_{3150}(263,\cdot)\)
\(\chi_{3150}(527,\cdot)\)
\(\chi_{3150}(653,\cdot)\)
\(\chi_{3150}(767,\cdot)\)
\(\chi_{3150}(1283,\cdot)\)
\(\chi_{3150}(1397,\cdot)\)
\(\chi_{3150}(1523,\cdot)\)
\(\chi_{3150}(1787,\cdot)\)
\(\chi_{3150}(1913,\cdot)\)
\(\chi_{3150}(2027,\cdot)\)
\(\chi_{3150}(2153,\cdot)\)
\(\chi_{3150}(2417,\cdot)\)
\(\chi_{3150}(2783,\cdot)\)
\(\chi_{3150}(3047,\cdot)\)
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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 ]
\((2801,127,451)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{9}{20}\right),e\left(\frac{2}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 3150 }(137, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{5}{12}\right)\) |
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sage:chi.jacobi_sum(n)
