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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1800, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,0,20,51]))
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pari:[g,chi] = znchar(Mod(247,1800))
\(\chi_{1800}(103,\cdot)\)
\(\chi_{1800}(223,\cdot)\)
\(\chi_{1800}(247,\cdot)\)
\(\chi_{1800}(367,\cdot)\)
\(\chi_{1800}(463,\cdot)\)
\(\chi_{1800}(583,\cdot)\)
\(\chi_{1800}(727,\cdot)\)
\(\chi_{1800}(823,\cdot)\)
\(\chi_{1800}(967,\cdot)\)
\(\chi_{1800}(1087,\cdot)\)
\(\chi_{1800}(1183,\cdot)\)
\(\chi_{1800}(1303,\cdot)\)
\(\chi_{1800}(1327,\cdot)\)
\(\chi_{1800}(1447,\cdot)\)
\(\chi_{1800}(1663,\cdot)\)
\(\chi_{1800}(1687,\cdot)\)
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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 ]
\((1351,901,1001,577)\) → \((-1,1,e\left(\frac{1}{3}\right),e\left(\frac{17}{20}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 1800 }(247, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{15}\right)\) |
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sage:chi.jacobi_sum(n)
