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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([0,280,189,150]))
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pari:[g,chi] = znchar(Mod(5137,44100))
\(\chi_{44100}(13,\cdot)\)
\(\chi_{44100}(517,\cdot)\)
\(\chi_{44100}(853,\cdot)\)
\(\chi_{44100}(1777,\cdot)\)
\(\chi_{44100}(2113,\cdot)\)
\(\chi_{44100}(2533,\cdot)\)
\(\chi_{44100}(2617,\cdot)\)
\(\chi_{44100}(3373,\cdot)\)
\(\chi_{44100}(3877,\cdot)\)
\(\chi_{44100}(4297,\cdot)\)
\(\chi_{44100}(4633,\cdot)\)
\(\chi_{44100}(5053,\cdot)\)
\(\chi_{44100}(5137,\cdot)\)
\(\chi_{44100}(6313,\cdot)\)
\(\chi_{44100}(6397,\cdot)\)
\(\chi_{44100}(6817,\cdot)\)
\(\chi_{44100}(7573,\cdot)\)
\(\chi_{44100}(8077,\cdot)\)
\(\chi_{44100}(8413,\cdot)\)
\(\chi_{44100}(8833,\cdot)\)
\(\chi_{44100}(9337,\cdot)\)
\(\chi_{44100}(9673,\cdot)\)
\(\chi_{44100}(10177,\cdot)\)
\(\chi_{44100}(10597,\cdot)\)
\(\chi_{44100}(10933,\cdot)\)
\(\chi_{44100}(11353,\cdot)\)
\(\chi_{44100}(11437,\cdot)\)
\(\chi_{44100}(12613,\cdot)\)
\(\chi_{44100}(12697,\cdot)\)
\(\chi_{44100}(13117,\cdot)\)
...
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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 ]
\((22051,34301,15877,9901)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{9}{20}\right),e\left(\frac{5}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 44100 }(5137, a) \) |
\(1\) | \(1\) | \(e\left(\frac{16}{105}\right)\) | \(e\left(\frac{281}{420}\right)\) | \(e\left(\frac{109}{140}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{359}{420}\right)\) | \(e\left(\frac{209}{210}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{67}{140}\right)\) | \(e\left(\frac{103}{210}\right)\) | \(e\left(\frac{47}{84}\right)\) |
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sage:chi.jacobi_sum(n)
