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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5616, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,9,26,0]))
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pari:[g,chi] = znchar(Mod(443,5616))
\(\chi_{5616}(131,\cdot)\)
\(\chi_{5616}(443,\cdot)\)
\(\chi_{5616}(1067,\cdot)\)
\(\chi_{5616}(1379,\cdot)\)
\(\chi_{5616}(2003,\cdot)\)
\(\chi_{5616}(2315,\cdot)\)
\(\chi_{5616}(2939,\cdot)\)
\(\chi_{5616}(3251,\cdot)\)
\(\chi_{5616}(3875,\cdot)\)
\(\chi_{5616}(4187,\cdot)\)
\(\chi_{5616}(4811,\cdot)\)
\(\chi_{5616}(5123,\cdot)\)
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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 ]
\((703,4213,2081,3889)\) → \((-1,i,e\left(\frac{13}{18}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 5616 }(443, a) \) |
\(1\) | \(1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) |
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sage:chi.jacobi_sum(n)
