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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8820, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,35,21,39]))
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pari:[g,chi] = znchar(Mod(1049,8820))
\(\chi_{8820}(209,\cdot)\)
\(\chi_{8820}(1049,\cdot)\)
\(\chi_{8820}(2309,\cdot)\)
\(\chi_{8820}(2729,\cdot)\)
\(\chi_{8820}(3569,\cdot)\)
\(\chi_{8820}(3989,\cdot)\)
\(\chi_{8820}(4829,\cdot)\)
\(\chi_{8820}(5249,\cdot)\)
\(\chi_{8820}(6089,\cdot)\)
\(\chi_{8820}(6509,\cdot)\)
\(\chi_{8820}(7769,\cdot)\)
\(\chi_{8820}(8609,\cdot)\)
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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 ]
\((4411,7841,7057,1081)\) → \((1,e\left(\frac{5}{6}\right),-1,e\left(\frac{13}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 8820 }(1049, a) \) |
\(1\) | \(1\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(-1\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{17}{42}\right)\) |
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sage:chi.jacobi_sum(n)
