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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4056, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([0,0,0,29]))
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pari:[g,chi] = znchar(Mod(1345,4056))
\(\chi_{4056}(97,\cdot)\)
\(\chi_{4056}(145,\cdot)\)
\(\chi_{4056}(193,\cdot)\)
\(\chi_{4056}(241,\cdot)\)
\(\chi_{4056}(409,\cdot)\)
\(\chi_{4056}(457,\cdot)\)
\(\chi_{4056}(505,\cdot)\)
\(\chi_{4056}(553,\cdot)\)
\(\chi_{4056}(721,\cdot)\)
\(\chi_{4056}(769,\cdot)\)
\(\chi_{4056}(817,\cdot)\)
\(\chi_{4056}(865,\cdot)\)
\(\chi_{4056}(1081,\cdot)\)
\(\chi_{4056}(1129,\cdot)\)
\(\chi_{4056}(1177,\cdot)\)
\(\chi_{4056}(1345,\cdot)\)
\(\chi_{4056}(1393,\cdot)\)
\(\chi_{4056}(1489,\cdot)\)
\(\chi_{4056}(1657,\cdot)\)
\(\chi_{4056}(1705,\cdot)\)
\(\chi_{4056}(1753,\cdot)\)
\(\chi_{4056}(1801,\cdot)\)
\(\chi_{4056}(1969,\cdot)\)
\(\chi_{4056}(2017,\cdot)\)
\(\chi_{4056}(2065,\cdot)\)
\(\chi_{4056}(2113,\cdot)\)
\(\chi_{4056}(2281,\cdot)\)
\(\chi_{4056}(2329,\cdot)\)
\(\chi_{4056}(2377,\cdot)\)
\(\chi_{4056}(2425,\cdot)\)
...
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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 ]
\((1015,2029,2705,3889)\) → \((1,1,1,e\left(\frac{29}{156}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 4056 }(1345, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{35}{52}\right)\) | \(e\left(\frac{139}{156}\right)\) | \(e\left(\frac{23}{156}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{47}{52}\right)\) | \(e\left(\frac{22}{39}\right)\) |
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sage:chi.jacobi_sum(n)
