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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2900, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,0,9]))
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pari:[g,chi] = znchar(Mod(251,2900))
\(\chi_{2900}(251,\cdot)\)
\(\chi_{2900}(351,\cdot)\)
\(\chi_{2900}(751,\cdot)\)
\(\chi_{2900}(851,\cdot)\)
\(\chi_{2900}(1551,\cdot)\)
\(\chi_{2900}(1651,\cdot)\)
\(\chi_{2900}(1751,\cdot)\)
\(\chi_{2900}(1951,\cdot)\)
\(\chi_{2900}(2051,\cdot)\)
\(\chi_{2900}(2251,\cdot)\)
\(\chi_{2900}(2351,\cdot)\)
\(\chi_{2900}(2451,\cdot)\)
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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 ]
\((1451,1277,901)\) → \((-1,1,e\left(\frac{9}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 2900 }(251, a) \) |
\(1\) | \(1\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(-i\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{9}{28}\right)\) |
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sage:chi.jacobi_sum(n)
