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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([15,15,12,20]))
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pari:[g,chi] = znchar(Mod(431,6300))
\(\chi_{6300}(431,\cdot)\)
\(\chi_{6300}(611,\cdot)\)
\(\chi_{6300}(1691,\cdot)\)
\(\chi_{6300}(1871,\cdot)\)
\(\chi_{6300}(3131,\cdot)\)
\(\chi_{6300}(4211,\cdot)\)
\(\chi_{6300}(4391,\cdot)\)
\(\chi_{6300}(5471,\cdot)\)
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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 ]
\((3151,2801,3277,3601)\) → \((-1,-1,e\left(\frac{2}{5}\right),e\left(\frac{2}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 6300 }(431, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(-1\) |
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sage:chi.jacobi_sum(n)
