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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(44100, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([0,14,7,24]))
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pari:[g,chi] = znchar(Mod(39257,44100))
\(\chi_{44100}(1457,\cdot)\)
\(\chi_{44100}(5993,\cdot)\)
\(\chi_{44100}(7757,\cdot)\)
\(\chi_{44100}(12293,\cdot)\)
\(\chi_{44100}(14057,\cdot)\)
\(\chi_{44100}(18593,\cdot)\)
\(\chi_{44100}(20357,\cdot)\)
\(\chi_{44100}(31193,\cdot)\)
\(\chi_{44100}(32957,\cdot)\)
\(\chi_{44100}(37493,\cdot)\)
\(\chi_{44100}(39257,\cdot)\)
\(\chi_{44100}(43793,\cdot)\)
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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 ]
\((22051,34301,15877,9901)\) → \((1,-1,i,e\left(\frac{6}{7}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 44100 }(39257, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(-1\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(1\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{25}{28}\right)\) |
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sage:chi.jacobi_sum(n)
