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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4830, base_ring=CyclotomicField(132))
M = H._module
chi = DirichletCharacter(H, M([0,99,44,126]))
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pari:[g,chi] = znchar(Mod(1003,4830))
\(\chi_{4830}(37,\cdot)\)
\(\chi_{4830}(67,\cdot)\)
\(\chi_{4830}(247,\cdot)\)
\(\chi_{4830}(373,\cdot)\)
\(\chi_{4830}(457,\cdot)\)
\(\chi_{4830}(613,\cdot)\)
\(\chi_{4830}(697,\cdot)\)
\(\chi_{4830}(793,\cdot)\)
\(\chi_{4830}(907,\cdot)\)
\(\chi_{4830}(1003,\cdot)\)
\(\chi_{4830}(1033,\cdot)\)
\(\chi_{4830}(1213,\cdot)\)
\(\chi_{4830}(1423,\cdot)\)
\(\chi_{4830}(1537,\cdot)\)
\(\chi_{4830}(1663,\cdot)\)
\(\chi_{4830}(1717,\cdot)\)
\(\chi_{4830}(1873,\cdot)\)
\(\chi_{4830}(2137,\cdot)\)
\(\chi_{4830}(2167,\cdot)\)
\(\chi_{4830}(2503,\cdot)\)
\(\chi_{4830}(2587,\cdot)\)
\(\chi_{4830}(2683,\cdot)\)
\(\chi_{4830}(2767,\cdot)\)
\(\chi_{4830}(2797,\cdot)\)
\(\chi_{4830}(2977,\cdot)\)
\(\chi_{4830}(3007,\cdot)\)
\(\chi_{4830}(3103,\cdot)\)
\(\chi_{4830}(3133,\cdot)\)
\(\chi_{4830}(3217,\cdot)\)
\(\chi_{4830}(3553,\cdot)\)
...
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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 ]
\((3221,967,2761,1891)\) → \((1,-i,e\left(\frac{1}{3}\right),e\left(\frac{21}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 4830 }(1003, a) \) |
\(1\) | \(1\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{101}{132}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{61}{132}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{5}{12}\right)\) |
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sage:chi.jacobi_sum(n)
