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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2940, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,21,21,25]))
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pari:[g,chi] = znchar(Mod(2369,2940))
\(\chi_{2940}(89,\cdot)\)
\(\chi_{2940}(269,\cdot)\)
\(\chi_{2940}(689,\cdot)\)
\(\chi_{2940}(929,\cdot)\)
\(\chi_{2940}(1349,\cdot)\)
\(\chi_{2940}(1529,\cdot)\)
\(\chi_{2940}(1769,\cdot)\)
\(\chi_{2940}(1949,\cdot)\)
\(\chi_{2940}(2189,\cdot)\)
\(\chi_{2940}(2369,\cdot)\)
\(\chi_{2940}(2609,\cdot)\)
\(\chi_{2940}(2789,\cdot)\)
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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 ]
\((1471,1961,1177,1081)\) → \((1,-1,-1,e\left(\frac{25}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 2940 }(2369, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) |
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sage:chi.jacobi_sum(n)
