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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1081, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([0,22]))
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pari:[g,chi] = znchar(Mod(921,1081))
\(\chi_{1081}(24,\cdot)\)
\(\chi_{1081}(162,\cdot)\)
\(\chi_{1081}(277,\cdot)\)
\(\chi_{1081}(300,\cdot)\)
\(\chi_{1081}(346,\cdot)\)
\(\chi_{1081}(392,\cdot)\)
\(\chi_{1081}(484,\cdot)\)
\(\chi_{1081}(507,\cdot)\)
\(\chi_{1081}(553,\cdot)\)
\(\chi_{1081}(576,\cdot)\)
\(\chi_{1081}(645,\cdot)\)
\(\chi_{1081}(714,\cdot)\)
\(\chi_{1081}(737,\cdot)\)
\(\chi_{1081}(760,\cdot)\)
\(\chi_{1081}(806,\cdot)\)
\(\chi_{1081}(852,\cdot)\)
\(\chi_{1081}(921,\cdot)\)
\(\chi_{1081}(944,\cdot)\)
\(\chi_{1081}(967,\cdot)\)
\(\chi_{1081}(990,\cdot)\)
\(\chi_{1081}(1036,\cdot)\)
\(\chi_{1081}(1059,\cdot)\)
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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 ]
\((189,898)\) → \((1,e\left(\frac{11}{23}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 1081 }(921, a) \) |
\(1\) | \(1\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) |
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sage:chi.jacobi_sum(n)
