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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([0,0,50,27]))
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pari:[g,chi] = znchar(Mod(1337,1800))
\(\chi_{1800}(113,\cdot)\)
\(\chi_{1800}(137,\cdot)\)
\(\chi_{1800}(353,\cdot)\)
\(\chi_{1800}(473,\cdot)\)
\(\chi_{1800}(497,\cdot)\)
\(\chi_{1800}(617,\cdot)\)
\(\chi_{1800}(713,\cdot)\)
\(\chi_{1800}(833,\cdot)\)
\(\chi_{1800}(977,\cdot)\)
\(\chi_{1800}(1073,\cdot)\)
\(\chi_{1800}(1217,\cdot)\)
\(\chi_{1800}(1337,\cdot)\)
\(\chi_{1800}(1433,\cdot)\)
\(\chi_{1800}(1553,\cdot)\)
\(\chi_{1800}(1577,\cdot)\)
\(\chi_{1800}(1697,\cdot)\)
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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 ]
\((1351,901,1001,577)\) → \((1,1,e\left(\frac{5}{6}\right),e\left(\frac{9}{20}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 1800 }(1337, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{29}{30}\right)\) |
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sage:chi.jacobi_sum(n)
