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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9075, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([0,22,14]))
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pari:[g,chi] = znchar(Mod(3316,9075))
\(\chi_{9075}(16,\cdot)\)
\(\chi_{9075}(256,\cdot)\)
\(\chi_{9075}(361,\cdot)\)
\(\chi_{9075}(796,\cdot)\)
\(\chi_{9075}(841,\cdot)\)
\(\chi_{9075}(1081,\cdot)\)
\(\chi_{9075}(1186,\cdot)\)
\(\chi_{9075}(1621,\cdot)\)
\(\chi_{9075}(1666,\cdot)\)
\(\chi_{9075}(1906,\cdot)\)
\(\chi_{9075}(2011,\cdot)\)
\(\chi_{9075}(2446,\cdot)\)
\(\chi_{9075}(2491,\cdot)\)
\(\chi_{9075}(2731,\cdot)\)
\(\chi_{9075}(2836,\cdot)\)
\(\chi_{9075}(3271,\cdot)\)
\(\chi_{9075}(3316,\cdot)\)
\(\chi_{9075}(3556,\cdot)\)
\(\chi_{9075}(3661,\cdot)\)
\(\chi_{9075}(4096,\cdot)\)
\(\chi_{9075}(4381,\cdot)\)
\(\chi_{9075}(4966,\cdot)\)
\(\chi_{9075}(5311,\cdot)\)
\(\chi_{9075}(5746,\cdot)\)
\(\chi_{9075}(5791,\cdot)\)
\(\chi_{9075}(6031,\cdot)\)
\(\chi_{9075}(6136,\cdot)\)
\(\chi_{9075}(6571,\cdot)\)
\(\chi_{9075}(6616,\cdot)\)
\(\chi_{9075}(6856,\cdot)\)
...
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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 ]
\((3026,727,5326)\) → \((1,e\left(\frac{1}{5}\right),e\left(\frac{7}{55}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
| \( \chi_{ 9075 }(3316, a) \) |
\(1\) | \(1\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{6}{55}\right)\) |
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sage:chi.jacobi_sum(n)
