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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3648, base_ring=CyclotomicField(144))
M = H._module
chi = DirichletCharacter(H, M([0,9,0,8]))
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pari:[g,chi] = znchar(Mod(325,3648))
\(\chi_{3648}(13,\cdot)\)
\(\chi_{3648}(109,\cdot)\)
\(\chi_{3648}(181,\cdot)\)
\(\chi_{3648}(205,\cdot)\)
\(\chi_{3648}(325,\cdot)\)
\(\chi_{3648}(421,\cdot)\)
\(\chi_{3648}(469,\cdot)\)
\(\chi_{3648}(565,\cdot)\)
\(\chi_{3648}(637,\cdot)\)
\(\chi_{3648}(661,\cdot)\)
\(\chi_{3648}(781,\cdot)\)
\(\chi_{3648}(877,\cdot)\)
\(\chi_{3648}(925,\cdot)\)
\(\chi_{3648}(1021,\cdot)\)
\(\chi_{3648}(1093,\cdot)\)
\(\chi_{3648}(1117,\cdot)\)
\(\chi_{3648}(1237,\cdot)\)
\(\chi_{3648}(1333,\cdot)\)
\(\chi_{3648}(1381,\cdot)\)
\(\chi_{3648}(1477,\cdot)\)
\(\chi_{3648}(1549,\cdot)\)
\(\chi_{3648}(1573,\cdot)\)
\(\chi_{3648}(1693,\cdot)\)
\(\chi_{3648}(1789,\cdot)\)
\(\chi_{3648}(1837,\cdot)\)
\(\chi_{3648}(1933,\cdot)\)
\(\chi_{3648}(2005,\cdot)\)
\(\chi_{3648}(2029,\cdot)\)
\(\chi_{3648}(2149,\cdot)\)
\(\chi_{3648}(2245,\cdot)\)
...
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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 ]
\((2623,2053,1217,1921)\) → \((1,e\left(\frac{1}{16}\right),1,e\left(\frac{1}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 3648 }(325, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{137}{144}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{31}{144}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{71}{72}\right)\) | \(e\left(\frac{65}{72}\right)\) | \(e\left(\frac{91}{144}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{131}{144}\right)\) |
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sage:chi.jacobi_sum(n)
