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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1870, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([20,16,65]))
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pari:[g,chi] = znchar(Mod(1797,1870))
\(\chi_{1870}(37,\cdot)\)
\(\chi_{1870}(97,\cdot)\)
\(\chi_{1870}(113,\cdot)\)
\(\chi_{1870}(163,\cdot)\)
\(\chi_{1870}(207,\cdot)\)
\(\chi_{1870}(267,\cdot)\)
\(\chi_{1870}(313,\cdot)\)
\(\chi_{1870}(333,\cdot)\)
\(\chi_{1870}(377,\cdot)\)
\(\chi_{1870}(533,\cdot)\)
\(\chi_{1870}(653,\cdot)\)
\(\chi_{1870}(823,\cdot)\)
\(\chi_{1870}(873,\cdot)\)
\(\chi_{1870}(993,\cdot)\)
\(\chi_{1870}(1017,\cdot)\)
\(\chi_{1870}(1043,\cdot)\)
\(\chi_{1870}(1127,\cdot)\)
\(\chi_{1870}(1213,\cdot)\)
\(\chi_{1870}(1303,\cdot)\)
\(\chi_{1870}(1357,\cdot)\)
\(\chi_{1870}(1457,\cdot)\)
\(\chi_{1870}(1467,\cdot)\)
\(\chi_{1870}(1523,\cdot)\)
\(\chi_{1870}(1527,\cdot)\)
\(\chi_{1870}(1567,\cdot)\)
\(\chi_{1870}(1637,\cdot)\)
\(\chi_{1870}(1643,\cdot)\)
\(\chi_{1870}(1697,\cdot)\)
\(\chi_{1870}(1797,\cdot)\)
\(\chi_{1870}(1807,\cdot)\)
...
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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 ]
\((1497,1531,1431)\) → \((i,e\left(\frac{1}{5}\right),e\left(\frac{13}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 1870 }(1797, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(-i\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{41}{80}\right)\) |
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sage:chi.jacobi_sum(n)
