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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2480, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,0,15,16]))
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pari:[g,chi] = znchar(Mod(369,2480))
\(\chi_{2480}(49,\cdot)\)
\(\chi_{2480}(289,\cdot)\)
\(\chi_{2480}(369,\cdot)\)
\(\chi_{2480}(609,\cdot)\)
\(\chi_{2480}(689,\cdot)\)
\(\chi_{2480}(1249,\cdot)\)
\(\chi_{2480}(1409,\cdot)\)
\(\chi_{2480}(1569,\cdot)\)
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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 ]
\((1551,1861,497,561)\) → \((1,1,-1,e\left(\frac{8}{15}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 2480 }(369, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) |
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sage:chi.jacobi_sum(n)
