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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5160, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,0,0,21,25]))
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pari:[g,chi] = znchar(Mod(1639,5160))
\(\chi_{5160}(319,\cdot)\)
\(\chi_{5160}(679,\cdot)\)
\(\chi_{5160}(1639,\cdot)\)
\(\chi_{5160}(1879,\cdot)\)
\(\chi_{5160}(2119,\cdot)\)
\(\chi_{5160}(2239,\cdot)\)
\(\chi_{5160}(2479,\cdot)\)
\(\chi_{5160}(2599,\cdot)\)
\(\chi_{5160}(3079,\cdot)\)
\(\chi_{5160}(3559,\cdot)\)
\(\chi_{5160}(4759,\cdot)\)
\(\chi_{5160}(4879,\cdot)\)
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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 ]
\((3871,2581,1721,3097,4561)\) → \((-1,1,1,-1,e\left(\frac{25}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 5160 }(1639, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{4}{7}\right)\) |
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sage:chi.jacobi_sum(n)
