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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([0,34,0,7]))
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pari:[g,chi] = znchar(Mod(21169,30345))
\(\chi_{30345}(64,\cdot)\)
\(\chi_{30345}(1534,\cdot)\)
\(\chi_{30345}(1849,\cdot)\)
\(\chi_{30345}(3319,\cdot)\)
\(\chi_{30345}(3634,\cdot)\)
\(\chi_{30345}(5104,\cdot)\)
\(\chi_{30345}(5419,\cdot)\)
\(\chi_{30345}(6889,\cdot)\)
\(\chi_{30345}(7204,\cdot)\)
\(\chi_{30345}(8674,\cdot)\)
\(\chi_{30345}(8989,\cdot)\)
\(\chi_{30345}(10459,\cdot)\)
\(\chi_{30345}(10774,\cdot)\)
\(\chi_{30345}(12244,\cdot)\)
\(\chi_{30345}(12559,\cdot)\)
\(\chi_{30345}(14029,\cdot)\)
\(\chi_{30345}(14344,\cdot)\)
\(\chi_{30345}(15814,\cdot)\)
\(\chi_{30345}(16129,\cdot)\)
\(\chi_{30345}(17599,\cdot)\)
\(\chi_{30345}(17914,\cdot)\)
\(\chi_{30345}(19384,\cdot)\)
\(\chi_{30345}(19699,\cdot)\)
\(\chi_{30345}(21169,\cdot)\)
\(\chi_{30345}(21484,\cdot)\)
\(\chi_{30345}(22954,\cdot)\)
\(\chi_{30345}(23269,\cdot)\)
\(\chi_{30345}(24739,\cdot)\)
\(\chi_{30345}(25054,\cdot)\)
\(\chi_{30345}(26524,\cdot)\)
...
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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 ]
\((20231,24277,4336,28036)\) → \((1,-1,1,e\left(\frac{7}{68}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(19\) | \(22\) | \(23\) | \(26\) |
| \( \chi_{ 30345 }(21169, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{25}{68}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{29}{68}\right)\) | \(e\left(\frac{31}{68}\right)\) | \(e\left(\frac{25}{34}\right)\) |
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sage:chi.jacobi_sum(n)
