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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([0,0,21,82]))
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pari:[g,chi] = znchar(Mod(33157,44100))
\(\chi_{44100}(1657,\cdot)\)
\(\chi_{44100}(2593,\cdot)\)
\(\chi_{44100}(4357,\cdot)\)
\(\chi_{44100}(8893,\cdot)\)
\(\chi_{44100}(10657,\cdot)\)
\(\chi_{44100}(12493,\cdot)\)
\(\chi_{44100}(14257,\cdot)\)
\(\chi_{44100}(15193,\cdot)\)
\(\chi_{44100}(16957,\cdot)\)
\(\chi_{44100}(18793,\cdot)\)
\(\chi_{44100}(20557,\cdot)\)
\(\chi_{44100}(25093,\cdot)\)
\(\chi_{44100}(26857,\cdot)\)
\(\chi_{44100}(27793,\cdot)\)
\(\chi_{44100}(29557,\cdot)\)
\(\chi_{44100}(31393,\cdot)\)
\(\chi_{44100}(33157,\cdot)\)
\(\chi_{44100}(34093,\cdot)\)
\(\chi_{44100}(35857,\cdot)\)
\(\chi_{44100}(37693,\cdot)\)
\(\chi_{44100}(39457,\cdot)\)
\(\chi_{44100}(40393,\cdot)\)
\(\chi_{44100}(42157,\cdot)\)
\(\chi_{44100}(43993,\cdot)\)
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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 ]
\((22051,34301,15877,9901)\) → \((1,1,i,e\left(\frac{41}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 44100 }(33157, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{17}{28}\right)\) |
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sage:chi.jacobi_sum(n)
