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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2700, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,50,3]))
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pari:[g,chi] = znchar(Mod(2177,2700))
\(\chi_{2700}(17,\cdot)\)
\(\chi_{2700}(197,\cdot)\)
\(\chi_{2700}(233,\cdot)\)
\(\chi_{2700}(413,\cdot)\)
\(\chi_{2700}(737,\cdot)\)
\(\chi_{2700}(773,\cdot)\)
\(\chi_{2700}(953,\cdot)\)
\(\chi_{2700}(1097,\cdot)\)
\(\chi_{2700}(1277,\cdot)\)
\(\chi_{2700}(1313,\cdot)\)
\(\chi_{2700}(1637,\cdot)\)
\(\chi_{2700}(1817,\cdot)\)
\(\chi_{2700}(1853,\cdot)\)
\(\chi_{2700}(2033,\cdot)\)
\(\chi_{2700}(2177,\cdot)\)
\(\chi_{2700}(2573,\cdot)\)
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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 ]
\((1351,1001,2377)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{1}{20}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 2700 }(2177, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{11}{30}\right)\) |
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sage:chi.jacobi_sum(n)
