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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(85600, base_ring=CyclotomicField(530))
M = H._module
chi = DirichletCharacter(H, M([265,0,106,105]))
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pari:[g,chi] = znchar(Mod(8191,85600))
\(\chi_{85600}(31,\cdot)\)
\(\chi_{85600}(191,\cdot)\)
\(\chi_{85600}(831,\cdot)\)
\(\chi_{85600}(991,\cdot)\)
\(\chi_{85600}(1471,\cdot)\)
\(\chi_{85600}(1631,\cdot)\)
\(\chi_{85600}(2111,\cdot)\)
\(\chi_{85600}(2271,\cdot)\)
\(\chi_{85600}(2431,\cdot)\)
\(\chi_{85600}(2911,\cdot)\)
\(\chi_{85600}(3231,\cdot)\)
\(\chi_{85600}(3391,\cdot)\)
\(\chi_{85600}(3711,\cdot)\)
\(\chi_{85600}(4031,\cdot)\)
\(\chi_{85600}(4191,\cdot)\)
\(\chi_{85600}(4511,\cdot)\)
\(\chi_{85600}(4671,\cdot)\)
\(\chi_{85600}(5311,\cdot)\)
\(\chi_{85600}(5631,\cdot)\)
\(\chi_{85600}(6271,\cdot)\)
\(\chi_{85600}(6911,\cdot)\)
\(\chi_{85600}(7391,\cdot)\)
\(\chi_{85600}(7711,\cdot)\)
\(\chi_{85600}(7871,\cdot)\)
\(\chi_{85600}(8031,\cdot)\)
\(\chi_{85600}(8191,\cdot)\)
\(\chi_{85600}(8511,\cdot)\)
\(\chi_{85600}(9311,\cdot)\)
\(\chi_{85600}(9471,\cdot)\)
\(\chi_{85600}(9791,\cdot)\)
...
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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 ]
\((26751,32101,82177,16801)\) → \((-1,1,e\left(\frac{1}{5}\right),e\left(\frac{21}{106}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 85600 }(8191, a) \) |
\(1\) | \(1\) | \(e\left(\frac{407}{530}\right)\) | \(e\left(\frac{1}{53}\right)\) | \(e\left(\frac{142}{265}\right)\) | \(e\left(\frac{31}{530}\right)\) | \(e\left(\frac{152}{265}\right)\) | \(e\left(\frac{183}{530}\right)\) | \(e\left(\frac{293}{530}\right)\) | \(e\left(\frac{417}{530}\right)\) | \(e\left(\frac{521}{530}\right)\) | \(e\left(\frac{161}{530}\right)\) |
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sage:chi.jacobi_sum(n)
