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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4830, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,11,22,4]))
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pari:[g,chi] = znchar(Mod(4717,4830))
\(\chi_{4830}(13,\cdot)\)
\(\chi_{4830}(223,\cdot)\)
\(\chi_{4830}(307,\cdot)\)
\(\chi_{4830}(853,\cdot)\)
\(\chi_{4830}(1273,\cdot)\)
\(\chi_{4830}(1567,\cdot)\)
\(\chi_{4830}(1777,\cdot)\)
\(\chi_{4830}(1987,\cdot)\)
\(\chi_{4830}(2197,\cdot)\)
\(\chi_{4830}(2533,\cdot)\)
\(\chi_{4830}(2617,\cdot)\)
\(\chi_{4830}(2743,\cdot)\)
\(\chi_{4830}(2953,\cdot)\)
\(\chi_{4830}(3163,\cdot)\)
\(\chi_{4830}(3247,\cdot)\)
\(\chi_{4830}(3583,\cdot)\)
\(\chi_{4830}(3877,\cdot)\)
\(\chi_{4830}(4087,\cdot)\)
\(\chi_{4830}(4213,\cdot)\)
\(\chi_{4830}(4717,\cdot)\)
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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 ]
\((3221,967,2761,1891)\) → \((1,i,-1,e\left(\frac{1}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 4830 }(4717, a) \) |
\(1\) | \(1\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(-i\) |
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sage:chi.jacobi_sum(n)
