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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(44100, base_ring=CyclotomicField(84))
M = H._module
chi = DirichletCharacter(H, M([0,70,21,12]))
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pari:[g,chi] = znchar(Mod(43457,44100))
\(\chi_{44100}(1793,\cdot)\)
\(\chi_{44100}(3557,\cdot)\)
\(\chi_{44100}(3893,\cdot)\)
\(\chi_{44100}(5657,\cdot)\)
\(\chi_{44100}(8093,\cdot)\)
\(\chi_{44100}(9857,\cdot)\)
\(\chi_{44100}(14393,\cdot)\)
\(\chi_{44100}(16157,\cdot)\)
\(\chi_{44100}(16493,\cdot)\)
\(\chi_{44100}(18257,\cdot)\)
\(\chi_{44100}(20693,\cdot)\)
\(\chi_{44100}(22457,\cdot)\)
\(\chi_{44100}(22793,\cdot)\)
\(\chi_{44100}(24557,\cdot)\)
\(\chi_{44100}(26993,\cdot)\)
\(\chi_{44100}(28757,\cdot)\)
\(\chi_{44100}(29093,\cdot)\)
\(\chi_{44100}(30857,\cdot)\)
\(\chi_{44100}(33293,\cdot)\)
\(\chi_{44100}(35057,\cdot)\)
\(\chi_{44100}(35393,\cdot)\)
\(\chi_{44100}(37157,\cdot)\)
\(\chi_{44100}(41693,\cdot)\)
\(\chi_{44100}(43457,\cdot)\)
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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 ]
\((22051,34301,15877,9901)\) → \((1,e\left(\frac{5}{6}\right),i,e\left(\frac{1}{7}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 44100 }(43457, a) \) |
\(1\) | \(1\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(-1\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{79}{84}\right)\) |
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sage:chi.jacobi_sum(n)
