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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(373527, base_ring=CyclotomicField(1470))
M = H._module
chi = DirichletCharacter(H, M([980,635,1029]))
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pari:[g,chi] = znchar(Mod(28717,373527))
\(\chi_{373527}(40,\cdot)\)
\(\chi_{373527}(481,\cdot)\)
\(\chi_{373527}(1564,\cdot)\)
\(\chi_{373527}(2635,\cdot)\)
\(\chi_{373527}(3379,\cdot)\)
\(\chi_{373527}(4450,\cdot)\)
\(\chi_{373527}(5848,\cdot)\)
\(\chi_{373527}(6289,\cdot)\)
\(\chi_{373527}(9187,\cdot)\)
\(\chi_{373527}(10258,\cdot)\)
\(\chi_{373527}(11002,\cdot)\)
\(\chi_{373527}(13471,\cdot)\)
\(\chi_{373527}(13912,\cdot)\)
\(\chi_{373527}(15286,\cdot)\)
\(\chi_{373527}(15727,\cdot)\)
\(\chi_{373527}(16810,\cdot)\)
\(\chi_{373527}(17881,\cdot)\)
\(\chi_{373527}(18625,\cdot)\)
\(\chi_{373527}(19696,\cdot)\)
\(\chi_{373527}(21094,\cdot)\)
\(\chi_{373527}(21535,\cdot)\)
\(\chi_{373527}(22909,\cdot)\)
\(\chi_{373527}(23350,\cdot)\)
\(\chi_{373527}(25504,\cdot)\)
\(\chi_{373527}(26248,\cdot)\)
\(\chi_{373527}(27319,\cdot)\)
\(\chi_{373527}(28717,\cdot)\)
\(\chi_{373527}(29158,\cdot)\)
\(\chi_{373527}(30532,\cdot)\)
\(\chi_{373527}(30973,\cdot)\)
...
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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 ]
\((290522,286408,126568)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{127}{294}\right),e\left(\frac{7}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
| \( \chi_{ 373527 }(28717, a) \) |
\(1\) | \(1\) | \(e\left(\frac{83}{490}\right)\) | \(e\left(\frac{83}{245}\right)\) | \(e\left(\frac{971}{1470}\right)\) | \(e\left(\frac{249}{490}\right)\) | \(e\left(\frac{122}{147}\right)\) | \(e\left(\frac{107}{735}\right)\) | \(e\left(\frac{166}{245}\right)\) | \(e\left(\frac{73}{735}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1469}{1470}\right)\) |
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sage:chi.jacobi_sum(n)
