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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([0,21,0,0,17]))
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pari:[g,chi] = znchar(Mod(2821,5160))
\(\chi_{5160}(61,\cdot)\)
\(\chi_{5160}(421,\cdot)\)
\(\chi_{5160}(1381,\cdot)\)
\(\chi_{5160}(1621,\cdot)\)
\(\chi_{5160}(1861,\cdot)\)
\(\chi_{5160}(1981,\cdot)\)
\(\chi_{5160}(2221,\cdot)\)
\(\chi_{5160}(2341,\cdot)\)
\(\chi_{5160}(2821,\cdot)\)
\(\chi_{5160}(3301,\cdot)\)
\(\chi_{5160}(4501,\cdot)\)
\(\chi_{5160}(4621,\cdot)\)
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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 ]
\((3871,2581,1721,3097,4561)\) → \((1,-1,1,1,e\left(\frac{17}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 5160 }(2821, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{3}{7}\right)\) |
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sage:chi.jacobi_sum(n)
