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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2057, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([24,5]))
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pari:[g,chi] = znchar(Mod(9,2057))
\(\chi_{2057}(9,\cdot)\)
\(\chi_{2057}(202,\cdot)\)
\(\chi_{2057}(366,\cdot)\)
\(\chi_{2057}(372,\cdot)\)
\(\chi_{2057}(444,\cdot)\)
\(\chi_{2057}(614,\cdot)\)
\(\chi_{2057}(729,\cdot)\)
\(\chi_{2057}(807,\cdot)\)
\(\chi_{2057}(971,\cdot)\)
\(\chi_{2057}(977,\cdot)\)
\(\chi_{2057}(995,\cdot)\)
\(\chi_{2057}(1334,\cdot)\)
\(\chi_{2057}(1358,\cdot)\)
\(\chi_{2057}(1600,\cdot)\)
\(\chi_{2057}(1896,\cdot)\)
\(\chi_{2057}(1963,\cdot)\)
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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 ]
\((970,122)\) → \((e\left(\frac{3}{5}\right),e\left(\frac{1}{8}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 2057 }(9, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) |
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sage:chi.jacobi_sum(n)
