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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2349, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([16,9]))
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pari:[g,chi] = znchar(Mod(1549,2349))
\(\chi_{2349}(46,\cdot)\)
\(\chi_{2349}(307,\cdot)\)
\(\chi_{2349}(505,\cdot)\)
\(\chi_{2349}(766,\cdot)\)
\(\chi_{2349}(829,\cdot)\)
\(\chi_{2349}(1090,\cdot)\)
\(\chi_{2349}(1288,\cdot)\)
\(\chi_{2349}(1549,\cdot)\)
\(\chi_{2349}(1612,\cdot)\)
\(\chi_{2349}(1873,\cdot)\)
\(\chi_{2349}(2071,\cdot)\)
\(\chi_{2349}(2332,\cdot)\)
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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 ]
\((407,1945)\) → \((e\left(\frac{4}{9}\right),i)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 2349 }(1549, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) |
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sage:chi.jacobi_sum(n)
