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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7800, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,0,30,18,25]))
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pari:[g,chi] = znchar(Mod(2489,7800))
\(\chi_{7800}(89,\cdot)\)
\(\chi_{7800}(929,\cdot)\)
\(\chi_{7800}(1289,\cdot)\)
\(\chi_{7800}(2009,\cdot)\)
\(\chi_{7800}(2489,\cdot)\)
\(\chi_{7800}(3209,\cdot)\)
\(\chi_{7800}(3569,\cdot)\)
\(\chi_{7800}(4409,\cdot)\)
\(\chi_{7800}(4769,\cdot)\)
\(\chi_{7800}(5129,\cdot)\)
\(\chi_{7800}(5609,\cdot)\)
\(\chi_{7800}(5969,\cdot)\)
\(\chi_{7800}(6329,\cdot)\)
\(\chi_{7800}(6689,\cdot)\)
\(\chi_{7800}(7169,\cdot)\)
\(\chi_{7800}(7529,\cdot)\)
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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 ]
\((1951,3901,5201,7177,4201)\) → \((1,1,-1,e\left(\frac{3}{10}\right),e\left(\frac{5}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 7800 }(2489, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{1}{3}\right)\) |
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sage:chi.jacobi_sum(n)
