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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([39,0,39,37]))
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pari:[g,chi] = znchar(Mod(95,4056))
\(\chi_{4056}(95,\cdot)\)
\(\chi_{4056}(335,\cdot)\)
\(\chi_{4056}(407,\cdot)\)
\(\chi_{4056}(647,\cdot)\)
\(\chi_{4056}(719,\cdot)\)
\(\chi_{4056}(959,\cdot)\)
\(\chi_{4056}(1031,\cdot)\)
\(\chi_{4056}(1271,\cdot)\)
\(\chi_{4056}(1343,\cdot)\)
\(\chi_{4056}(1583,\cdot)\)
\(\chi_{4056}(1655,\cdot)\)
\(\chi_{4056}(1895,\cdot)\)
\(\chi_{4056}(1967,\cdot)\)
\(\chi_{4056}(2207,\cdot)\)
\(\chi_{4056}(2279,\cdot)\)
\(\chi_{4056}(2519,\cdot)\)
\(\chi_{4056}(2591,\cdot)\)
\(\chi_{4056}(2831,\cdot)\)
\(\chi_{4056}(2903,\cdot)\)
\(\chi_{4056}(3143,\cdot)\)
\(\chi_{4056}(3215,\cdot)\)
\(\chi_{4056}(3455,\cdot)\)
\(\chi_{4056}(3767,\cdot)\)
\(\chi_{4056}(3839,\cdot)\)
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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 ]
\((1015,2029,2705,3889)\) → \((-1,1,-1,e\left(\frac{37}{78}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 4056 }(95, a) \) |
\(1\) | \(1\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{1}{39}\right)\) |
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sage:chi.jacobi_sum(n)
