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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(373527, base_ring=CyclotomicField(294))
M = H._module
chi = DirichletCharacter(H, M([0,1,0]))
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pari:[g,chi] = znchar(Mod(286408,373527))
\(\chi_{373527}(1090,\cdot)\)
\(\chi_{373527}(4357,\cdot)\)
\(\chi_{373527}(8713,\cdot)\)
\(\chi_{373527}(11980,\cdot)\)
\(\chi_{373527}(19603,\cdot)\)
\(\chi_{373527}(23959,\cdot)\)
\(\chi_{373527}(31582,\cdot)\)
\(\chi_{373527}(34849,\cdot)\)
\(\chi_{373527}(39205,\cdot)\)
\(\chi_{373527}(42472,\cdot)\)
\(\chi_{373527}(46828,\cdot)\)
\(\chi_{373527}(50095,\cdot)\)
\(\chi_{373527}(54451,\cdot)\)
\(\chi_{373527}(57718,\cdot)\)
\(\chi_{373527}(62074,\cdot)\)
\(\chi_{373527}(65341,\cdot)\)
\(\chi_{373527}(72964,\cdot)\)
\(\chi_{373527}(77320,\cdot)\)
\(\chi_{373527}(84943,\cdot)\)
\(\chi_{373527}(88210,\cdot)\)
\(\chi_{373527}(92566,\cdot)\)
\(\chi_{373527}(95833,\cdot)\)
\(\chi_{373527}(100189,\cdot)\)
\(\chi_{373527}(103456,\cdot)\)
\(\chi_{373527}(107812,\cdot)\)
\(\chi_{373527}(111079,\cdot)\)
\(\chi_{373527}(115435,\cdot)\)
\(\chi_{373527}(118702,\cdot)\)
\(\chi_{373527}(126325,\cdot)\)
\(\chi_{373527}(130681,\cdot)\)
...
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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 ]
\((290522,286408,126568)\) → \((1,e\left(\frac{1}{294}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
| \( \chi_{ 373527 }(286408, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{97}{147}\right)\) | \(e\left(\frac{47}{147}\right)\) | \(e\left(\frac{29}{294}\right)\) | \(e\left(\frac{48}{49}\right)\) | \(e\left(\frac{223}{294}\right)\) | \(e\left(\frac{95}{98}\right)\) | \(e\left(\frac{94}{147}\right)\) | \(e\left(\frac{25}{294}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{41}{98}\right)\) |
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sage:chi.jacobi_sum(n)
