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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(15800, base_ring=CyclotomicField(780))
M = H._module
chi = DirichletCharacter(H, M([0,0,741,350]))
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pari:[g,chi] = znchar(Mod(513,15800))
\(\chi_{15800}(113,\cdot)\)
\(\chi_{15800}(153,\cdot)\)
\(\chi_{15800}(217,\cdot)\)
\(\chi_{15800}(233,\cdot)\)
\(\chi_{15800}(297,\cdot)\)
\(\chi_{15800}(353,\cdot)\)
\(\chi_{15800}(513,\cdot)\)
\(\chi_{15800}(537,\cdot)\)
\(\chi_{15800}(777,\cdot)\)
\(\chi_{15800}(833,\cdot)\)
\(\chi_{15800}(897,\cdot)\)
\(\chi_{15800}(937,\cdot)\)
\(\chi_{15800}(977,\cdot)\)
\(\chi_{15800}(1033,\cdot)\)
\(\chi_{15800}(1097,\cdot)\)
\(\chi_{15800}(1113,\cdot)\)
\(\chi_{15800}(1153,\cdot)\)
\(\chi_{15800}(1233,\cdot)\)
\(\chi_{15800}(1377,\cdot)\)
\(\chi_{15800}(1417,\cdot)\)
\(\chi_{15800}(1497,\cdot)\)
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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 ]
\((3951,7901,11377,12801)\) → \((1,1,e\left(\frac{19}{20}\right),e\left(\frac{35}{78}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 15800 }(513, a) \) |
\(1\) | \(1\) | \(e\left(\frac{77}{780}\right)\) | \(e\left(\frac{83}{156}\right)\) | \(e\left(\frac{77}{390}\right)\) | \(e\left(\frac{139}{195}\right)\) | \(e\left(\frac{239}{780}\right)\) | \(e\left(\frac{201}{260}\right)\) | \(e\left(\frac{179}{390}\right)\) | \(e\left(\frac{41}{65}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{77}{260}\right)\) |
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sage:chi.jacobi_sum(n)
