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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4056, base_ring=CyclotomicField(52))
M = H._module
chi = DirichletCharacter(H, M([0,0,0,37]))
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pari:[g,chi] = znchar(Mod(1633,4056))
\(\chi_{4056}(73,\cdot)\)
\(\chi_{4056}(265,\cdot)\)
\(\chi_{4056}(385,\cdot)\)
\(\chi_{4056}(697,\cdot)\)
\(\chi_{4056}(889,\cdot)\)
\(\chi_{4056}(1009,\cdot)\)
\(\chi_{4056}(1201,\cdot)\)
\(\chi_{4056}(1321,\cdot)\)
\(\chi_{4056}(1513,\cdot)\)
\(\chi_{4056}(1633,\cdot)\)
\(\chi_{4056}(1825,\cdot)\)
\(\chi_{4056}(1945,\cdot)\)
\(\chi_{4056}(2137,\cdot)\)
\(\chi_{4056}(2257,\cdot)\)
\(\chi_{4056}(2449,\cdot)\)
\(\chi_{4056}(2569,\cdot)\)
\(\chi_{4056}(2761,\cdot)\)
\(\chi_{4056}(2881,\cdot)\)
\(\chi_{4056}(3073,\cdot)\)
\(\chi_{4056}(3193,\cdot)\)
\(\chi_{4056}(3385,\cdot)\)
\(\chi_{4056}(3505,\cdot)\)
\(\chi_{4056}(3697,\cdot)\)
\(\chi_{4056}(4009,\cdot)\)
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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 ]
\((1015,2029,2705,3889)\) → \((1,1,1,e\left(\frac{37}{52}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 4056 }(1633, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{21}{52}\right)\) | \(e\left(\frac{7}{52}\right)\) | \(e\left(\frac{15}{52}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(i\) | \(-1\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{49}{52}\right)\) | \(e\left(\frac{7}{13}\right)\) |
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sage:chi.jacobi_sum(n)
