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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(373527, base_ring=CyclotomicField(16170))
M = H._module
chi = DirichletCharacter(H, M([8085,550,15582]))
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pari:[g,chi] = znchar(Mod(53,373527))
\(\chi_{373527}(53,\cdot)\)
\(\chi_{373527}(170,\cdot)\)
\(\chi_{373527}(179,\cdot)\)
\(\chi_{373527}(368,\cdot)\)
\(\chi_{373527}(548,\cdot)\)
\(\chi_{373527}(620,\cdot)\)
\(\chi_{373527}(674,\cdot)\)
\(\chi_{373527}(746,\cdot)\)
\(\chi_{373527}(872,\cdot)\)
\(\chi_{373527}(1061,\cdot)\)
\(\chi_{373527}(1115,\cdot)\)
\(\chi_{373527}(1241,\cdot)\)
\(\chi_{373527}(1313,\cdot)\)
\(\chi_{373527}(1367,\cdot)\)
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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 ]
\((290522,286408,126568)\) → \((-1,e\left(\frac{5}{147}\right),e\left(\frac{53}{55}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
| \( \chi_{ 373527 }(53, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1007}{16170}\right)\) | \(e\left(\frac{1007}{8085}\right)\) | \(e\left(\frac{12863}{16170}\right)\) | \(e\left(\frac{1007}{5390}\right)\) | \(e\left(\frac{1387}{1617}\right)\) | \(e\left(\frac{57}{2695}\right)\) | \(e\left(\frac{2014}{8085}\right)\) | \(e\left(\frac{9193}{16170}\right)\) | \(e\left(\frac{52}{165}\right)\) | \(e\left(\frac{4959}{5390}\right)\) |
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sage:chi.jacobi_sum(n)
