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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(30345, base_ring=CyclotomicField(68))
M = H._module
chi = DirichletCharacter(H, M([0,51,34,44]))
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pari:[g,chi] = znchar(Mod(8518,30345))
\(\chi_{30345}(307,\cdot)\)
\(\chi_{30345}(1378,\cdot)\)
\(\chi_{30345}(2092,\cdot)\)
\(\chi_{30345}(3163,\cdot)\)
\(\chi_{30345}(3877,\cdot)\)
\(\chi_{30345}(4948,\cdot)\)
\(\chi_{30345}(5662,\cdot)\)
\(\chi_{30345}(6733,\cdot)\)
\(\chi_{30345}(7447,\cdot)\)
\(\chi_{30345}(8518,\cdot)\)
\(\chi_{30345}(9232,\cdot)\)
\(\chi_{30345}(10303,\cdot)\)
\(\chi_{30345}(11017,\cdot)\)
\(\chi_{30345}(12088,\cdot)\)
\(\chi_{30345}(12802,\cdot)\)
\(\chi_{30345}(14587,\cdot)\)
\(\chi_{30345}(15658,\cdot)\)
\(\chi_{30345}(16372,\cdot)\)
\(\chi_{30345}(17443,\cdot)\)
\(\chi_{30345}(18157,\cdot)\)
\(\chi_{30345}(19228,\cdot)\)
\(\chi_{30345}(21013,\cdot)\)
\(\chi_{30345}(21727,\cdot)\)
\(\chi_{30345}(22798,\cdot)\)
\(\chi_{30345}(23512,\cdot)\)
\(\chi_{30345}(24583,\cdot)\)
\(\chi_{30345}(25297,\cdot)\)
\(\chi_{30345}(26368,\cdot)\)
\(\chi_{30345}(27082,\cdot)\)
\(\chi_{30345}(28153,\cdot)\)
...
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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 ]
\((20231,24277,4336,28036)\) → \((1,-i,-1,e\left(\frac{11}{17}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(19\) | \(22\) | \(23\) | \(26\) |
| \( \chi_{ 30345 }(8518, a) \) |
\(1\) | \(1\) | \(e\left(\frac{47}{68}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{5}{68}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{39}{68}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{39}{68}\right)\) | \(e\left(\frac{37}{68}\right)\) | \(e\left(\frac{9}{34}\right)\) |
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sage:chi.jacobi_sum(n)
