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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2156, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,4,0]))
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pari:[g,chi] = znchar(Mod(375,2156))
\(\chi_{2156}(23,\cdot)\)
\(\chi_{2156}(331,\cdot)\)
\(\chi_{2156}(375,\cdot)\)
\(\chi_{2156}(639,\cdot)\)
\(\chi_{2156}(683,\cdot)\)
\(\chi_{2156}(947,\cdot)\)
\(\chi_{2156}(991,\cdot)\)
\(\chi_{2156}(1299,\cdot)\)
\(\chi_{2156}(1563,\cdot)\)
\(\chi_{2156}(1607,\cdot)\)
\(\chi_{2156}(1871,\cdot)\)
\(\chi_{2156}(1915,\cdot)\)
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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 ]
\((1079,1277,981)\) → \((-1,e\left(\frac{2}{21}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 2156 }(375, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{11}{14}\right)\) |
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sage:chi.jacobi_sum(n)
