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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4020, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,0,0,47]))
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pari:[g,chi] = znchar(Mod(31,4020))
\(\chi_{4020}(31,\cdot)\)
\(\chi_{4020}(331,\cdot)\)
\(\chi_{4020}(631,\cdot)\)
\(\chi_{4020}(811,\cdot)\)
\(\chi_{4020}(1051,\cdot)\)
\(\chi_{4020}(1171,\cdot)\)
\(\chi_{4020}(1291,\cdot)\)
\(\chi_{4020}(1351,\cdot)\)
\(\chi_{4020}(1531,\cdot)\)
\(\chi_{4020}(1591,\cdot)\)
\(\chi_{4020}(2071,\cdot)\)
\(\chi_{4020}(2491,\cdot)\)
\(\chi_{4020}(2731,\cdot)\)
\(\chi_{4020}(2791,\cdot)\)
\(\chi_{4020}(3151,\cdot)\)
\(\chi_{4020}(3331,\cdot)\)
\(\chi_{4020}(3391,\cdot)\)
\(\chi_{4020}(3451,\cdot)\)
\(\chi_{4020}(3571,\cdot)\)
\(\chi_{4020}(3631,\cdot)\)
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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 ]
\((2011,2681,3217,1141)\) → \((-1,1,1,e\left(\frac{47}{66}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 4020 }(31, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{49}{66}\right)\) |
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sage:chi.jacobi_sum(n)
