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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9900, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,20,57,6]))
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pari:[g,chi] = znchar(Mod(13,9900))
\(\chi_{9900}(13,\cdot)\)
\(\chi_{9900}(733,\cdot)\)
\(\chi_{9900}(1777,\cdot)\)
\(\chi_{9900}(2173,\cdot)\)
\(\chi_{9900}(2317,\cdot)\)
\(\chi_{9900}(3253,\cdot)\)
\(\chi_{9900}(4237,\cdot)\)
\(\chi_{9900}(5497,\cdot)\)
\(\chi_{9900}(6613,\cdot)\)
\(\chi_{9900}(7333,\cdot)\)
\(\chi_{9900}(7537,\cdot)\)
\(\chi_{9900}(8377,\cdot)\)
\(\chi_{9900}(8773,\cdot)\)
\(\chi_{9900}(8797,\cdot)\)
\(\chi_{9900}(8917,\cdot)\)
\(\chi_{9900}(9853,\cdot)\)
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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 ]
\((4951,5501,2377,4501)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{19}{20}\right),e\left(\frac{1}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 9900 }(13, a) \) |
\(1\) | \(1\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(i\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(-i\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{1}{12}\right)\) |
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sage:chi.jacobi_sum(n)
