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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(880, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,15,0,18]))
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pari:[g,chi] = znchar(Mod(61,880))
\(\chi_{880}(61,\cdot)\)
\(\chi_{880}(101,\cdot)\)
\(\chi_{880}(261,\cdot)\)
\(\chi_{880}(381,\cdot)\)
\(\chi_{880}(501,\cdot)\)
\(\chi_{880}(541,\cdot)\)
\(\chi_{880}(701,\cdot)\)
\(\chi_{880}(821,\cdot)\)
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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 ]
\((111,661,177,321)\) → \((1,-i,1,e\left(\frac{9}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 880 }(61, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(i\) | \(-1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) |
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sage:chi.jacobi_sum(n)
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sage:chi.gauss_sum(a)
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pari:znchargauss(g,chi,a)
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sage:chi.jacobi_sum(n)
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sage:chi.kloosterman_sum(a,b)
