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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4176, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,21,14,6]))
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pari:[g,chi] = znchar(Mod(35,4176))
\(\chi_{4176}(35,\cdot)\)
\(\chi_{4176}(179,\cdot)\)
\(\chi_{4176}(323,\cdot)\)
\(\chi_{4176}(1115,\cdot)\)
\(\chi_{4176}(1691,\cdot)\)
\(\chi_{4176}(1907,\cdot)\)
\(\chi_{4176}(2123,\cdot)\)
\(\chi_{4176}(2267,\cdot)\)
\(\chi_{4176}(2411,\cdot)\)
\(\chi_{4176}(3203,\cdot)\)
\(\chi_{4176}(3779,\cdot)\)
\(\chi_{4176}(3995,\cdot)\)
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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 ]
\((1567,1045,929,4033)\) → \((-1,-i,-1,e\left(\frac{3}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(31\) | \(35\) |
| \( \chi_{ 4176 }(35, a) \) |
\(1\) | \(1\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(1\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{15}{28}\right)\) |
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sage:chi.jacobi_sum(n)
