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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1060, base_ring=CyclotomicField(52))
M = H._module
chi = DirichletCharacter(H, M([26,0,41]))
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pari:[g,chi] = znchar(Mod(251,1060))
\(\chi_{1060}(31,\cdot)\)
\(\chi_{1060}(51,\cdot)\)
\(\chi_{1060}(71,\cdot)\)
\(\chi_{1060}(111,\cdot)\)
\(\chi_{1060}(151,\cdot)\)
\(\chi_{1060}(171,\cdot)\)
\(\chi_{1060}(191,\cdot)\)
\(\chi_{1060}(231,\cdot)\)
\(\chi_{1060}(251,\cdot)\)
\(\chi_{1060}(291,\cdot)\)
\(\chi_{1060}(351,\cdot)\)
\(\chi_{1060}(391,\cdot)\)
\(\chi_{1060}(451,\cdot)\)
\(\chi_{1060}(491,\cdot)\)
\(\chi_{1060}(511,\cdot)\)
\(\chi_{1060}(551,\cdot)\)
\(\chi_{1060}(571,\cdot)\)
\(\chi_{1060}(591,\cdot)\)
\(\chi_{1060}(631,\cdot)\)
\(\chi_{1060}(671,\cdot)\)
\(\chi_{1060}(691,\cdot)\)
\(\chi_{1060}(711,\cdot)\)
\(\chi_{1060}(851,\cdot)\)
\(\chi_{1060}(951,\cdot)\)
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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 ]
\((531,637,161)\) → \((-1,1,e\left(\frac{41}{52}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 1060 }(251, a) \) |
\(1\) | \(1\) | \(e\left(\frac{47}{52}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{35}{52}\right)\) | \(e\left(\frac{23}{52}\right)\) | \(i\) | \(e\left(\frac{37}{52}\right)\) |
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sage:chi.jacobi_sum(n)
