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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2856, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,24,0,40,3]))
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pari:[g,chi] = znchar(Mod(2077,2856))
\(\chi_{2856}(61,\cdot)\)
\(\chi_{2856}(397,\cdot)\)
\(\chi_{2856}(997,\cdot)\)
\(\chi_{2856}(1333,\cdot)\)
\(\chi_{2856}(1405,\cdot)\)
\(\chi_{2856}(1501,\cdot)\)
\(\chi_{2856}(1669,\cdot)\)
\(\chi_{2856}(1741,\cdot)\)
\(\chi_{2856}(1909,\cdot)\)
\(\chi_{2856}(2077,\cdot)\)
\(\chi_{2856}(2173,\cdot)\)
\(\chi_{2856}(2341,\cdot)\)
\(\chi_{2856}(2509,\cdot)\)
\(\chi_{2856}(2581,\cdot)\)
\(\chi_{2856}(2749,\cdot)\)
\(\chi_{2856}(2845,\cdot)\)
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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 ]
\((2143,1429,953,409,2689)\) → \((1,-1,1,e\left(\frac{5}{6}\right),e\left(\frac{1}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 2856 }(2077, a) \) |
\(1\) | \(1\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(i\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{3}{16}\right)\) |
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sage:chi.jacobi_sum(n)
