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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1875, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([0,37]))
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pari:[g,chi] = znchar(Mod(424,1875))
\(\chi_{1875}(49,\cdot)\)
\(\chi_{1875}(199,\cdot)\)
\(\chi_{1875}(274,\cdot)\)
\(\chi_{1875}(349,\cdot)\)
\(\chi_{1875}(424,\cdot)\)
\(\chi_{1875}(574,\cdot)\)
\(\chi_{1875}(649,\cdot)\)
\(\chi_{1875}(724,\cdot)\)
\(\chi_{1875}(799,\cdot)\)
\(\chi_{1875}(949,\cdot)\)
\(\chi_{1875}(1024,\cdot)\)
\(\chi_{1875}(1099,\cdot)\)
\(\chi_{1875}(1174,\cdot)\)
\(\chi_{1875}(1324,\cdot)\)
\(\chi_{1875}(1399,\cdot)\)
\(\chi_{1875}(1474,\cdot)\)
\(\chi_{1875}(1549,\cdot)\)
\(\chi_{1875}(1699,\cdot)\)
\(\chi_{1875}(1774,\cdot)\)
\(\chi_{1875}(1849,\cdot)\)
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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 ]
\((626,1252)\) → \((1,e\left(\frac{37}{50}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1875 }(424, a) \) |
\(1\) | \(1\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{8}{25}\right)\) |
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sage:chi.jacobi_sum(n)
