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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(740, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,9,7]))
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pari:[g,chi] = znchar(Mod(17,740))
\(\chi_{740}(17,\cdot)\)
\(\chi_{740}(113,\cdot)\)
\(\chi_{740}(257,\cdot)\)
\(\chi_{740}(353,\cdot)\)
\(\chi_{740}(457,\cdot)\)
\(\chi_{740}(513,\cdot)\)
\(\chi_{740}(533,\cdot)\)
\(\chi_{740}(537,\cdot)\)
\(\chi_{740}(573,\cdot)\)
\(\chi_{740}(577,\cdot)\)
\(\chi_{740}(597,\cdot)\)
\(\chi_{740}(653,\cdot)\)
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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 ]
\((371,297,261)\) → \((1,i,e\left(\frac{7}{36}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 740 }(17, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{12}\right)\) |
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sage:chi.jacobi_sum(n)
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sage:chi.gauss_sum(a)
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pari:znchargauss(g,chi,a)
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sage:chi.jacobi_sum(n)
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sage:chi.kloosterman_sum(a,b)
