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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(44100, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,20,27,30]))
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pari:[g,chi] = znchar(Mod(3037,44100))
\(\chi_{44100}(97,\cdot)\)
\(\chi_{44100}(1273,\cdot)\)
\(\chi_{44100}(3037,\cdot)\)
\(\chi_{44100}(7153,\cdot)\)
\(\chi_{44100}(8917,\cdot)\)
\(\chi_{44100}(15973,\cdot)\)
\(\chi_{44100}(17737,\cdot)\)
\(\chi_{44100}(18913,\cdot)\)
\(\chi_{44100}(20677,\cdot)\)
\(\chi_{44100}(27733,\cdot)\)
\(\chi_{44100}(29497,\cdot)\)
\(\chi_{44100}(33613,\cdot)\)
\(\chi_{44100}(35377,\cdot)\)
\(\chi_{44100}(36553,\cdot)\)
\(\chi_{44100}(38317,\cdot)\)
\(\chi_{44100}(42433,\cdot)\)
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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 ]
\((22051,34301,15877,9901)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{9}{20}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 44100 }(3037, a) \) |
\(1\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{12}\right)\) |
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sage:chi.jacobi_sum(n)
