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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7056, base_ring=CyclotomicField(84))
M = H._module
chi = DirichletCharacter(H, M([0,63,42,46]))
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pari:[g,chi] = znchar(Mod(1853,7056))
\(\chi_{7056}(269,\cdot)\)
\(\chi_{7056}(341,\cdot)\)
\(\chi_{7056}(773,\cdot)\)
\(\chi_{7056}(845,\cdot)\)
\(\chi_{7056}(1277,\cdot)\)
\(\chi_{7056}(1349,\cdot)\)
\(\chi_{7056}(1781,\cdot)\)
\(\chi_{7056}(1853,\cdot)\)
\(\chi_{7056}(2357,\cdot)\)
\(\chi_{7056}(2789,\cdot)\)
\(\chi_{7056}(3293,\cdot)\)
\(\chi_{7056}(3365,\cdot)\)
\(\chi_{7056}(3797,\cdot)\)
\(\chi_{7056}(3869,\cdot)\)
\(\chi_{7056}(4301,\cdot)\)
\(\chi_{7056}(4373,\cdot)\)
\(\chi_{7056}(4805,\cdot)\)
\(\chi_{7056}(4877,\cdot)\)
\(\chi_{7056}(5309,\cdot)\)
\(\chi_{7056}(5381,\cdot)\)
\(\chi_{7056}(5885,\cdot)\)
\(\chi_{7056}(6317,\cdot)\)
\(\chi_{7056}(6821,\cdot)\)
\(\chi_{7056}(6893,\cdot)\)
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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 ]
\((6175,1765,785,4609)\) → \((1,-i,-1,e\left(\frac{23}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 7056 }(1853, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{23}{84}\right)\) |
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sage:chi.jacobi_sum(n)
