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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([0,21,0,11]))
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pari:[g,chi] = znchar(Mod(61,1960))
\(\chi_{1960}(61,\cdot)\)
\(\chi_{1960}(101,\cdot)\)
\(\chi_{1960}(341,\cdot)\)
\(\chi_{1960}(381,\cdot)\)
\(\chi_{1960}(621,\cdot)\)
\(\chi_{1960}(661,\cdot)\)
\(\chi_{1960}(941,\cdot)\)
\(\chi_{1960}(1181,\cdot)\)
\(\chi_{1960}(1221,\cdot)\)
\(\chi_{1960}(1461,\cdot)\)
\(\chi_{1960}(1741,\cdot)\)
\(\chi_{1960}(1781,\cdot)\)
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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 ]
\((1471,981,1177,1081)\) → \((1,-1,1,e\left(\frac{11}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 1960 }(61, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{5}{6}\right)\) |
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sage:chi.jacobi_sum(n)
