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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4056, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([39,0,0,56]))
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pari:[g,chi] = znchar(Mod(55,4056))
\(\chi_{4056}(55,\cdot)\)
\(\chi_{4056}(295,\cdot)\)
\(\chi_{4056}(367,\cdot)\)
\(\chi_{4056}(607,\cdot)\)
\(\chi_{4056}(679,\cdot)\)
\(\chi_{4056}(919,\cdot)\)
\(\chi_{4056}(1231,\cdot)\)
\(\chi_{4056}(1303,\cdot)\)
\(\chi_{4056}(1615,\cdot)\)
\(\chi_{4056}(1855,\cdot)\)
\(\chi_{4056}(1927,\cdot)\)
\(\chi_{4056}(2167,\cdot)\)
\(\chi_{4056}(2239,\cdot)\)
\(\chi_{4056}(2479,\cdot)\)
\(\chi_{4056}(2551,\cdot)\)
\(\chi_{4056}(2791,\cdot)\)
\(\chi_{4056}(2863,\cdot)\)
\(\chi_{4056}(3103,\cdot)\)
\(\chi_{4056}(3175,\cdot)\)
\(\chi_{4056}(3415,\cdot)\)
\(\chi_{4056}(3487,\cdot)\)
\(\chi_{4056}(3727,\cdot)\)
\(\chi_{4056}(3799,\cdot)\)
\(\chi_{4056}(4039,\cdot)\)
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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 ]
\((1015,2029,2705,3889)\) → \((-1,1,1,e\left(\frac{28}{39}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 4056 }(55, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{61}{78}\right)\) |
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sage:chi.jacobi_sum(n)
