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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4020, base_ring=CyclotomicField(132))
M = H._module
chi = DirichletCharacter(H, M([66,0,33,122]))
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pari:[g,chi] = znchar(Mod(1987,4020))
\(\chi_{4020}(7,\cdot)\)
\(\chi_{4020}(247,\cdot)\)
\(\chi_{4020}(367,\cdot)\)
\(\chi_{4020}(463,\cdot)\)
\(\chi_{4020}(487,\cdot)\)
\(\chi_{4020}(547,\cdot)\)
\(\chi_{4020}(727,\cdot)\)
\(\chi_{4020}(787,\cdot)\)
\(\chi_{4020}(883,\cdot)\)
\(\chi_{4020}(1123,\cdot)\)
\(\chi_{4020}(1183,\cdot)\)
\(\chi_{4020}(1267,\cdot)\)
\(\chi_{4020}(1543,\cdot)\)
\(\chi_{4020}(1687,\cdot)\)
\(\chi_{4020}(1723,\cdot)\)
\(\chi_{4020}(1783,\cdot)\)
\(\chi_{4020}(1843,\cdot)\)
\(\chi_{4020}(1927,\cdot)\)
\(\chi_{4020}(1963,\cdot)\)
\(\chi_{4020}(1987,\cdot)\)
\(\chi_{4020}(2023,\cdot)\)
\(\chi_{4020}(2347,\cdot)\)
\(\chi_{4020}(2443,\cdot)\)
\(\chi_{4020}(2527,\cdot)\)
\(\chi_{4020}(2587,\cdot)\)
\(\chi_{4020}(2647,\cdot)\)
\(\chi_{4020}(2743,\cdot)\)
\(\chi_{4020}(2767,\cdot)\)
\(\chi_{4020}(2827,\cdot)\)
\(\chi_{4020}(3043,\cdot)\)
...
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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 ]
\((2011,2681,3217,1141)\) → \((-1,1,i,e\left(\frac{61}{66}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 4020 }(1987, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{132}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{41}{132}\right)\) | \(e\left(\frac{53}{132}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{17}{132}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{65}{66}\right)\) |
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sage:chi.jacobi_sum(n)
