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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6300, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,10,21,10]))
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pari:[g,chi] = znchar(Mod(4909,6300))
\(\chi_{6300}(529,\cdot)\)
\(\chi_{6300}(1129,\cdot)\)
\(\chi_{6300}(1789,\cdot)\)
\(\chi_{6300}(2389,\cdot)\)
\(\chi_{6300}(4309,\cdot)\)
\(\chi_{6300}(4909,\cdot)\)
\(\chi_{6300}(5569,\cdot)\)
\(\chi_{6300}(6169,\cdot)\)
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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 ]
\((3151,2801,3277,3601)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{7}{10}\right),e\left(\frac{1}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 6300 }(4909, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{5}{6}\right)\) |
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sage:chi.jacobi_sum(n)
