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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(44100, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([0,35,42,155]))
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pari:[g,chi] = znchar(Mod(5141,44100))
\(\chi_{44100}(761,\cdot)\)
\(\chi_{44100}(1361,\cdot)\)
\(\chi_{44100}(2021,\cdot)\)
\(\chi_{44100}(2621,\cdot)\)
\(\chi_{44100}(3281,\cdot)\)
\(\chi_{44100}(3881,\cdot)\)
\(\chi_{44100}(4541,\cdot)\)
\(\chi_{44100}(5141,\cdot)\)
\(\chi_{44100}(7061,\cdot)\)
\(\chi_{44100}(7661,\cdot)\)
\(\chi_{44100}(8321,\cdot)\)
\(\chi_{44100}(8921,\cdot)\)
\(\chi_{44100}(9581,\cdot)\)
\(\chi_{44100}(10181,\cdot)\)
\(\chi_{44100}(10841,\cdot)\)
\(\chi_{44100}(11441,\cdot)\)
\(\chi_{44100}(13361,\cdot)\)
\(\chi_{44100}(13961,\cdot)\)
\(\chi_{44100}(15881,\cdot)\)
\(\chi_{44100}(16481,\cdot)\)
\(\chi_{44100}(17141,\cdot)\)
\(\chi_{44100}(17741,\cdot)\)
\(\chi_{44100}(19661,\cdot)\)
\(\chi_{44100}(20261,\cdot)\)
\(\chi_{44100}(20921,\cdot)\)
\(\chi_{44100}(21521,\cdot)\)
\(\chi_{44100}(22181,\cdot)\)
\(\chi_{44100}(22781,\cdot)\)
\(\chi_{44100}(25961,\cdot)\)
\(\chi_{44100}(26561,\cdot)\)
...
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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 ]
\((22051,34301,15877,9901)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{1}{5}\right),e\left(\frac{31}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 44100 }(5141, a) \) |
\(1\) | \(1\) | \(e\left(\frac{187}{210}\right)\) | \(e\left(\frac{103}{210}\right)\) | \(e\left(\frac{58}{105}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{17}{210}\right)\) | \(e\left(\frac{179}{210}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{44}{105}\right)\) | \(e\left(\frac{74}{105}\right)\) | \(e\left(\frac{2}{21}\right)\) |
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sage:chi.jacobi_sum(n)
