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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([210,210,357,410]))
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pari:[g,chi] = znchar(Mod(4247,44100))
\(\chi_{44100}(467,\cdot)\)
\(\chi_{44100}(647,\cdot)\)
\(\chi_{44100}(1223,\cdot)\)
\(\chi_{44100}(1727,\cdot)\)
\(\chi_{44100}(2483,\cdot)\)
\(\chi_{44100}(2663,\cdot)\)
\(\chi_{44100}(2987,\cdot)\)
\(\chi_{44100}(3923,\cdot)\)
\(\chi_{44100}(4247,\cdot)\)
\(\chi_{44100}(4427,\cdot)\)
\(\chi_{44100}(5003,\cdot)\)
\(\chi_{44100}(5183,\cdot)\)
\(\chi_{44100}(5687,\cdot)\)
\(\chi_{44100}(6263,\cdot)\)
\(\chi_{44100}(6767,\cdot)\)
\(\chi_{44100}(6947,\cdot)\)
\(\chi_{44100}(7523,\cdot)\)
\(\chi_{44100}(7703,\cdot)\)
\(\chi_{44100}(8027,\cdot)\)
\(\chi_{44100}(8783,\cdot)\)
\(\chi_{44100}(8963,\cdot)\)
\(\chi_{44100}(9287,\cdot)\)
\(\chi_{44100}(9467,\cdot)\)
\(\chi_{44100}(10547,\cdot)\)
\(\chi_{44100}(10727,\cdot)\)
\(\chi_{44100}(11303,\cdot)\)
\(\chi_{44100}(11483,\cdot)\)
\(\chi_{44100}(13067,\cdot)\)
\(\chi_{44100}(13247,\cdot)\)
\(\chi_{44100}(13823,\cdot)\)
...
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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 ]
\((22051,34301,15877,9901)\) → \((-1,-1,e\left(\frac{17}{20}\right),e\left(\frac{41}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 44100 }(4247, a) \) |
\(1\) | \(1\) | \(e\left(\frac{68}{105}\right)\) | \(e\left(\frac{51}{140}\right)\) | \(e\left(\frac{401}{420}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{187}{420}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{373}{420}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{3}{28}\right)\) |
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sage:chi.jacobi_sum(n)
