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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1240, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,0,45,44]))
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pari:[g,chi] = znchar(Mod(1223,1240))
\(\chi_{1240}(7,\cdot)\)
\(\chi_{1240}(103,\cdot)\)
\(\chi_{1240}(143,\cdot)\)
\(\chi_{1240}(183,\cdot)\)
\(\chi_{1240}(423,\cdot)\)
\(\chi_{1240}(503,\cdot)\)
\(\chi_{1240}(567,\cdot)\)
\(\chi_{1240}(607,\cdot)\)
\(\chi_{1240}(727,\cdot)\)
\(\chi_{1240}(847,\cdot)\)
\(\chi_{1240}(887,\cdot)\)
\(\chi_{1240}(927,\cdot)\)
\(\chi_{1240}(1063,\cdot)\)
\(\chi_{1240}(1103,\cdot)\)
\(\chi_{1240}(1167,\cdot)\)
\(\chi_{1240}(1223,\cdot)\)
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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 ]
\((311,621,497,561)\) → \((-1,1,-i,e\left(\frac{11}{15}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 1240 }(1223, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) |
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sage:chi.jacobi_sum(n)
