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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4600, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,22,11,32]))
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pari:[g,chi] = znchar(Mod(1107,4600))
\(\chi_{4600}(243,\cdot)\)
\(\chi_{4600}(307,\cdot)\)
\(\chi_{4600}(443,\cdot)\)
\(\chi_{4600}(1043,\cdot)\)
\(\chi_{4600}(1107,\cdot)\)
\(\chi_{4600}(1507,\cdot)\)
\(\chi_{4600}(1843,\cdot)\)
\(\chi_{4600}(2243,\cdot)\)
\(\chi_{4600}(2707,\cdot)\)
\(\chi_{4600}(2907,\cdot)\)
\(\chi_{4600}(3107,\cdot)\)
\(\chi_{4600}(3307,\cdot)\)
\(\chi_{4600}(3443,\cdot)\)
\(\chi_{4600}(3643,\cdot)\)
\(\chi_{4600}(3707,\cdot)\)
\(\chi_{4600}(3843,\cdot)\)
\(\chi_{4600}(4043,\cdot)\)
\(\chi_{4600}(4107,\cdot)\)
\(\chi_{4600}(4307,\cdot)\)
\(\chi_{4600}(4443,\cdot)\)
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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 ]
\((1151,2301,2577,1201)\) → \((-1,-1,i,e\left(\frac{8}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 4600 }(1107, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{1}{11}\right)\) |
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sage:chi.jacobi_sum(n)
