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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1976, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,0,33,10]))
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pari:[g,chi] = znchar(Mod(241,1976))
\(\chi_{1976}(41,\cdot)\)
\(\chi_{1976}(193,\cdot)\)
\(\chi_{1976}(241,\cdot)\)
\(\chi_{1976}(249,\cdot)\)
\(\chi_{1976}(401,\cdot)\)
\(\chi_{1976}(409,\cdot)\)
\(\chi_{1976}(865,\cdot)\)
\(\chi_{1976}(1345,\cdot)\)
\(\chi_{1976}(1705,\cdot)\)
\(\chi_{1976}(1761,\cdot)\)
\(\chi_{1976}(1801,\cdot)\)
\(\chi_{1976}(1857,\cdot)\)
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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 ]
\((495,989,457,1769)\) → \((1,1,e\left(\frac{11}{12}\right),e\left(\frac{5}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 1976 }(241, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(-i\) | \(e\left(\frac{5}{9}\right)\) | \(-i\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{7}{18}\right)\) |
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sage:chi.jacobi_sum(n)
