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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3528, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,21,0,29]))
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pari:[g,chi] = znchar(Mod(397,3528))
\(\chi_{3528}(397,\cdot)\)
\(\chi_{3528}(829,\cdot)\)
\(\chi_{3528}(1333,\cdot)\)
\(\chi_{3528}(1405,\cdot)\)
\(\chi_{3528}(1837,\cdot)\)
\(\chi_{3528}(1909,\cdot)\)
\(\chi_{3528}(2341,\cdot)\)
\(\chi_{3528}(2413,\cdot)\)
\(\chi_{3528}(2845,\cdot)\)
\(\chi_{3528}(2917,\cdot)\)
\(\chi_{3528}(3349,\cdot)\)
\(\chi_{3528}(3421,\cdot)\)
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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 ]
\((2647,1765,785,1081)\) → \((1,-1,1,e\left(\frac{29}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 3528 }(397, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{25}{42}\right)\) |
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sage:chi.jacobi_sum(n)
