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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1960, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,21,0,40]))
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pari:[g,chi] = znchar(Mod(11,1960))
\(\chi_{1960}(11,\cdot)\)
\(\chi_{1960}(51,\cdot)\)
\(\chi_{1960}(291,\cdot)\)
\(\chi_{1960}(331,\cdot)\)
\(\chi_{1960}(571,\cdot)\)
\(\chi_{1960}(611,\cdot)\)
\(\chi_{1960}(891,\cdot)\)
\(\chi_{1960}(1131,\cdot)\)
\(\chi_{1960}(1171,\cdot)\)
\(\chi_{1960}(1411,\cdot)\)
\(\chi_{1960}(1691,\cdot)\)
\(\chi_{1960}(1731,\cdot)\)
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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 ]
\((1471,981,1177,1081)\) → \((-1,-1,1,e\left(\frac{20}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 1960 }(11, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{1}{6}\right)\) |
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sage:chi.jacobi_sum(n)
