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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4056, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,0,0,16]))
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pari:[g,chi] = znchar(Mod(313,4056))
\(\chi_{4056}(313,\cdot)\)
\(\chi_{4056}(625,\cdot)\)
\(\chi_{4056}(937,\cdot)\)
\(\chi_{4056}(1249,\cdot)\)
\(\chi_{4056}(1561,\cdot)\)
\(\chi_{4056}(1873,\cdot)\)
\(\chi_{4056}(2185,\cdot)\)
\(\chi_{4056}(2497,\cdot)\)
\(\chi_{4056}(2809,\cdot)\)
\(\chi_{4056}(3121,\cdot)\)
\(\chi_{4056}(3433,\cdot)\)
\(\chi_{4056}(3745,\cdot)\)
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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 ]
\((1015,2029,2705,3889)\) → \((1,1,1,e\left(\frac{8}{13}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 4056 }(313, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(1\) | \(1\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) |
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sage:chi.jacobi_sum(n)
