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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(11025, base_ring=CyclotomicField(84))
M = H._module
chi = DirichletCharacter(H, M([56,63,38]))
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pari:[g,chi] = znchar(Mod(8368,11025))
\(\chi_{11025}(418,\cdot)\)
\(\chi_{11025}(493,\cdot)\)
\(\chi_{11025}(682,\cdot)\)
\(\chi_{11025}(1993,\cdot)\)
\(\chi_{11025}(2068,\cdot)\)
\(\chi_{11025}(2182,\cdot)\)
\(\chi_{11025}(2257,\cdot)\)
\(\chi_{11025}(3568,\cdot)\)
\(\chi_{11025}(3643,\cdot)\)
\(\chi_{11025}(3757,\cdot)\)
\(\chi_{11025}(3832,\cdot)\)
\(\chi_{11025}(5143,\cdot)\)
\(\chi_{11025}(5218,\cdot)\)
\(\chi_{11025}(5332,\cdot)\)
\(\chi_{11025}(5407,\cdot)\)
\(\chi_{11025}(6718,\cdot)\)
\(\chi_{11025}(6907,\cdot)\)
\(\chi_{11025}(6982,\cdot)\)
\(\chi_{11025}(8293,\cdot)\)
\(\chi_{11025}(8368,\cdot)\)
\(\chi_{11025}(8482,\cdot)\)
\(\chi_{11025}(9943,\cdot)\)
\(\chi_{11025}(10057,\cdot)\)
\(\chi_{11025}(10132,\cdot)\)
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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 ]
\((1226,4852,9901)\) → \((e\left(\frac{2}{3}\right),-i,e\left(\frac{19}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
| \( \chi_{ 11025 }(8368, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{65}{84}\right)\) |
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sage:chi.jacobi_sum(n)
