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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2890, base_ring=CyclotomicField(68))
M = H._module
chi = DirichletCharacter(H, M([0,5]))
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pari:[g,chi] = znchar(Mod(2291,2890))
\(\chi_{2890}(21,\cdot)\)
\(\chi_{2890}(81,\cdot)\)
\(\chi_{2890}(191,\cdot)\)
\(\chi_{2890}(361,\cdot)\)
\(\chi_{2890}(421,\cdot)\)
\(\chi_{2890}(531,\cdot)\)
\(\chi_{2890}(591,\cdot)\)
\(\chi_{2890}(701,\cdot)\)
\(\chi_{2890}(761,\cdot)\)
\(\chi_{2890}(871,\cdot)\)
\(\chi_{2890}(931,\cdot)\)
\(\chi_{2890}(1041,\cdot)\)
\(\chi_{2890}(1101,\cdot)\)
\(\chi_{2890}(1211,\cdot)\)
\(\chi_{2890}(1271,\cdot)\)
\(\chi_{2890}(1381,\cdot)\)
\(\chi_{2890}(1441,\cdot)\)
\(\chi_{2890}(1551,\cdot)\)
\(\chi_{2890}(1611,\cdot)\)
\(\chi_{2890}(1721,\cdot)\)
\(\chi_{2890}(1781,\cdot)\)
\(\chi_{2890}(1891,\cdot)\)
\(\chi_{2890}(1951,\cdot)\)
\(\chi_{2890}(2121,\cdot)\)
\(\chi_{2890}(2231,\cdot)\)
\(\chi_{2890}(2291,\cdot)\)
\(\chi_{2890}(2401,\cdot)\)
\(\chi_{2890}(2461,\cdot)\)
\(\chi_{2890}(2571,\cdot)\)
\(\chi_{2890}(2631,\cdot)\)
...
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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 ]
\((1157,581)\) → \((1,e\left(\frac{5}{68}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 2890 }(2291, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{68}\right)\) | \(e\left(\frac{27}{68}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{47}{68}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{27}{68}\right)\) | \(e\left(\frac{15}{68}\right)\) | \(e\left(\frac{13}{68}\right)\) |
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sage:chi.jacobi_sum(n)
