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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([10,51,50]))
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pari:[g,chi] = znchar(Mod(47,3150))
\(\chi_{3150}(47,\cdot)\)
\(\chi_{3150}(173,\cdot)\)
\(\chi_{3150}(437,\cdot)\)
\(\chi_{3150}(563,\cdot)\)
\(\chi_{3150}(677,\cdot)\)
\(\chi_{3150}(803,\cdot)\)
\(\chi_{3150}(1067,\cdot)\)
\(\chi_{3150}(1433,\cdot)\)
\(\chi_{3150}(1697,\cdot)\)
\(\chi_{3150}(1823,\cdot)\)
\(\chi_{3150}(1937,\cdot)\)
\(\chi_{3150}(2063,\cdot)\)
\(\chi_{3150}(2327,\cdot)\)
\(\chi_{3150}(2453,\cdot)\)
\(\chi_{3150}(2567,\cdot)\)
\(\chi_{3150}(3083,\cdot)\)
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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 ]
\((2801,127,451)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{17}{20}\right),e\left(\frac{5}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 3150 }(47, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{5}{12}\right)\) |
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sage:chi.jacobi_sum(n)
