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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(44100, base_ring=CyclotomicField(420))
M = H._module
chi = DirichletCharacter(H, M([0,0,147,170]))
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pari:[g,chi] = znchar(Mod(1153,44100))
\(\chi_{44100}(73,\cdot)\)
\(\chi_{44100}(397,\cdot)\)
\(\chi_{44100}(577,\cdot)\)
\(\chi_{44100}(1153,\cdot)\)
\(\chi_{44100}(1333,\cdot)\)
\(\chi_{44100}(1837,\cdot)\)
\(\chi_{44100}(2413,\cdot)\)
\(\chi_{44100}(2917,\cdot)\)
\(\chi_{44100}(3097,\cdot)\)
\(\chi_{44100}(3673,\cdot)\)
\(\chi_{44100}(4177,\cdot)\)
\(\chi_{44100}(4933,\cdot)\)
\(\chi_{44100}(5113,\cdot)\)
\(\chi_{44100}(5437,\cdot)\)
\(\chi_{44100}(6373,\cdot)\)
\(\chi_{44100}(6697,\cdot)\)
\(\chi_{44100}(6877,\cdot)\)
\(\chi_{44100}(7453,\cdot)\)
\(\chi_{44100}(7633,\cdot)\)
\(\chi_{44100}(8137,\cdot)\)
\(\chi_{44100}(8713,\cdot)\)
\(\chi_{44100}(9217,\cdot)\)
\(\chi_{44100}(9397,\cdot)\)
\(\chi_{44100}(9973,\cdot)\)
\(\chi_{44100}(10153,\cdot)\)
\(\chi_{44100}(10477,\cdot)\)
\(\chi_{44100}(11233,\cdot)\)
\(\chi_{44100}(11413,\cdot)\)
\(\chi_{44100}(11737,\cdot)\)
\(\chi_{44100}(11917,\cdot)\)
...
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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 ]
\((22051,34301,15877,9901)\) → \((1,1,e\left(\frac{7}{20}\right),e\left(\frac{17}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 44100 }(1153, a) \) |
\(1\) | \(1\) | \(e\left(\frac{83}{105}\right)\) | \(e\left(\frac{1}{140}\right)\) | \(e\left(\frac{281}{420}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{97}{420}\right)\) | \(e\left(\frac{69}{70}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{43}{420}\right)\) | \(e\left(\frac{33}{70}\right)\) | \(e\left(\frac{19}{28}\right)\) |
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sage:chi.jacobi_sum(n)
