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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(508288, base_ring=CyclotomicField(2736))
M = H._module
chi = DirichletCharacter(H, M([1368,1197,0,1168]))
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pari:[g,chi] = znchar(Mod(23,508288))
\(\chi_{508288}(23,\cdot)\)
\(\chi_{508288}(199,\cdot)\)
\(\chi_{508288}(727,\cdot)\)
\(\chi_{508288}(1431,\cdot)\)
\(\chi_{508288}(1783,\cdot)\)
\(\chi_{508288}(2487,\cdot)\)
\(\chi_{508288}(3367,\cdot)\)
\(\chi_{508288}(3543,\cdot)\)
\(\chi_{508288}(4071,\cdot)\)
\(\chi_{508288}(4775,\cdot)\)
\(\chi_{508288}(5127,\cdot)\)
\(\chi_{508288}(5831,\cdot)\)
\(\chi_{508288}(6711,\cdot)\)
\(\chi_{508288}(7415,\cdot)\)
\(\chi_{508288}(8119,\cdot)\)
\(\chi_{508288}(8471,\cdot)\)
\(\chi_{508288}(9175,\cdot)\)
\(\chi_{508288}(10055,\cdot)\)
\(\chi_{508288}(10231,\cdot)\)
\(\chi_{508288}(10759,\cdot)\)
\(\chi_{508288}(11463,\cdot)\)
\(\chi_{508288}(11815,\cdot)\)
\(\chi_{508288}(13399,\cdot)\)
\(\chi_{508288}(13575,\cdot)\)
\(\chi_{508288}(14103,\cdot)\)
\(\chi_{508288}(14807,\cdot)\)
\(\chi_{508288}(15159,\cdot)\)
\(\chi_{508288}(15863,\cdot)\)
\(\chi_{508288}(16743,\cdot)\)
\(\chi_{508288}(16919,\cdot)\)
...
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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 ]
\((166783,174725,323457,14081)\) → \((-1,e\left(\frac{7}{16}\right),1,e\left(\frac{73}{171}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 508288 }(23, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{415}{2736}\right)\) | \(e\left(\frac{1309}{2736}\right)\) | \(e\left(\frac{415}{456}\right)\) | \(e\left(\frac{415}{1368}\right)\) | \(e\left(\frac{35}{2736}\right)\) | \(e\left(\frac{431}{684}\right)\) | \(e\left(\frac{499}{684}\right)\) | \(e\left(\frac{169}{2736}\right)\) | \(e\left(\frac{1303}{1368}\right)\) | \(e\left(\frac{1309}{1368}\right)\) |
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sage:chi.jacobi_sum(n)
