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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(42237, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([0,19,12]))
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pari:[g,chi] = znchar(Mod(19837,42237))
\(\chi_{42237}(2053,\cdot)\)
\(\chi_{42237}(4276,\cdot)\)
\(\chi_{42237}(8722,\cdot)\)
\(\chi_{42237}(10945,\cdot)\)
\(\chi_{42237}(13168,\cdot)\)
\(\chi_{42237}(15391,\cdot)\)
\(\chi_{42237}(17614,\cdot)\)
\(\chi_{42237}(19837,\cdot)\)
\(\chi_{42237}(22060,\cdot)\)
\(\chi_{42237}(24283,\cdot)\)
\(\chi_{42237}(26506,\cdot)\)
\(\chi_{42237}(28729,\cdot)\)
\(\chi_{42237}(30952,\cdot)\)
\(\chi_{42237}(33175,\cdot)\)
\(\chi_{42237}(35398,\cdot)\)
\(\chi_{42237}(37621,\cdot)\)
\(\chi_{42237}(39844,\cdot)\)
\(\chi_{42237}(42067,\cdot)\)
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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 ]
\((32852,38989,12637)\) → \((1,-1,e\left(\frac{6}{19}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 42237 }(19837, a) \) |
\(1\) | \(1\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{7}{19}\right)\) |
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sage:chi.jacobi_sum(n)
