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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([0,140,168,100]))
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pari:[g,chi] = znchar(Mod(19861,44100))
\(\chi_{44100}(2221,\cdot)\)
\(\chi_{44100}(2461,\cdot)\)
\(\chi_{44100}(3481,\cdot)\)
\(\chi_{44100}(3721,\cdot)\)
\(\chi_{44100}(4741,\cdot)\)
\(\chi_{44100}(4981,\cdot)\)
\(\chi_{44100}(7261,\cdot)\)
\(\chi_{44100}(8521,\cdot)\)
\(\chi_{44100}(8761,\cdot)\)
\(\chi_{44100}(10021,\cdot)\)
\(\chi_{44100}(11041,\cdot)\)
\(\chi_{44100}(11281,\cdot)\)
\(\chi_{44100}(12541,\cdot)\)
\(\chi_{44100}(13561,\cdot)\)
\(\chi_{44100}(14821,\cdot)\)
\(\chi_{44100}(16081,\cdot)\)
\(\chi_{44100}(16321,\cdot)\)
\(\chi_{44100}(17341,\cdot)\)
\(\chi_{44100}(17581,\cdot)\)
\(\chi_{44100}(18841,\cdot)\)
\(\chi_{44100}(19861,\cdot)\)
\(\chi_{44100}(21121,\cdot)\)
\(\chi_{44100}(21361,\cdot)\)
\(\chi_{44100}(22381,\cdot)\)
\(\chi_{44100}(22621,\cdot)\)
\(\chi_{44100}(23641,\cdot)\)
\(\chi_{44100}(25141,\cdot)\)
\(\chi_{44100}(26161,\cdot)\)
\(\chi_{44100}(27661,\cdot)\)
\(\chi_{44100}(28681,\cdot)\)
...
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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 ]
\((22051,34301,15877,9901)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{4}{5}\right),e\left(\frac{10}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 44100 }(19861, a) \) |
\(1\) | \(1\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{26}{105}\right)\) | \(e\left(\frac{32}{105}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{88}{105}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{46}{105}\right)\) | \(e\left(\frac{71}{105}\right)\) | \(e\left(\frac{11}{21}\right)\) |
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sage:chi.jacobi_sum(n)
