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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1045, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([9,0,10]))
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pari:[g,chi] = znchar(Mod(507,1045))
\(\chi_{1045}(67,\cdot)\)
\(\chi_{1045}(78,\cdot)\)
\(\chi_{1045}(243,\cdot)\)
\(\chi_{1045}(287,\cdot)\)
\(\chi_{1045}(298,\cdot)\)
\(\chi_{1045}(452,\cdot)\)
\(\chi_{1045}(507,\cdot)\)
\(\chi_{1045}(573,\cdot)\)
\(\chi_{1045}(782,\cdot)\)
\(\chi_{1045}(793,\cdot)\)
\(\chi_{1045}(903,\cdot)\)
\(\chi_{1045}(1002,\cdot)\)
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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 ]
\((837,761,496)\) → \((i,1,e\left(\frac{5}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 1045 }(507, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{4}{9}\right)\) |
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sage:chi.jacobi_sum(n)
