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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2580, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,21,0,13]))
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pari:[g,chi] = znchar(Mod(2291,2580))
\(\chi_{2580}(71,\cdot)\)
\(\chi_{2580}(191,\cdot)\)
\(\chi_{2580}(491,\cdot)\)
\(\chi_{2580}(671,\cdot)\)
\(\chi_{2580}(851,\cdot)\)
\(\chi_{2580}(1151,\cdot)\)
\(\chi_{2580}(1811,\cdot)\)
\(\chi_{2580}(2051,\cdot)\)
\(\chi_{2580}(2291,\cdot)\)
\(\chi_{2580}(2351,\cdot)\)
\(\chi_{2580}(2411,\cdot)\)
\(\chi_{2580}(2471,\cdot)\)
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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 ]
\((1291,1721,517,1981)\) → \((-1,-1,1,e\left(\frac{13}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 2580 }(2291, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{14}\right)\) |
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sage:chi.jacobi_sum(n)
