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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5160, base_ring=CyclotomicField(84))
M = H._module
chi = DirichletCharacter(H, M([0,0,42,21,80]))
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pari:[g,chi] = znchar(Mod(497,5160))
\(\chi_{5160}(17,\cdot)\)
\(\chi_{5160}(353,\cdot)\)
\(\chi_{5160}(497,\cdot)\)
\(\chi_{5160}(617,\cdot)\)
\(\chi_{5160}(713,\cdot)\)
\(\chi_{5160}(857,\cdot)\)
\(\chi_{5160}(977,\cdot)\)
\(\chi_{5160}(1217,\cdot)\)
\(\chi_{5160}(1313,\cdot)\)
\(\chi_{5160}(1433,\cdot)\)
\(\chi_{5160}(1457,\cdot)\)
\(\chi_{5160}(2417,\cdot)\)
\(\chi_{5160}(2633,\cdot)\)
\(\chi_{5160}(2777,\cdot)\)
\(\chi_{5160}(3113,\cdot)\)
\(\chi_{5160}(3377,\cdot)\)
\(\chi_{5160}(3497,\cdot)\)
\(\chi_{5160}(3593,\cdot)\)
\(\chi_{5160}(3713,\cdot)\)
\(\chi_{5160}(3953,\cdot)\)
\(\chi_{5160}(4073,\cdot)\)
\(\chi_{5160}(4313,\cdot)\)
\(\chi_{5160}(4553,\cdot)\)
\(\chi_{5160}(4697,\cdot)\)
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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 ]
\((3871,2581,1721,3097,4561)\) → \((1,1,-1,i,e\left(\frac{20}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 5160 }(497, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{3}{14}\right)\) |
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sage:chi.jacobi_sum(n)
