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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(14700, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,21,0,37]))
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pari:[g,chi] = znchar(Mod(8501,14700))
\(\chi_{14700}(101,\cdot)\)
\(\chi_{14700}(1601,\cdot)\)
\(\chi_{14700}(2201,\cdot)\)
\(\chi_{14700}(3701,\cdot)\)
\(\chi_{14700}(4301,\cdot)\)
\(\chi_{14700}(7901,\cdot)\)
\(\chi_{14700}(8501,\cdot)\)
\(\chi_{14700}(10001,\cdot)\)
\(\chi_{14700}(10601,\cdot)\)
\(\chi_{14700}(12101,\cdot)\)
\(\chi_{14700}(12701,\cdot)\)
\(\chi_{14700}(14201,\cdot)\)
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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 ]
\((7351,4901,1177,9901)\) → \((1,-1,1,e\left(\frac{37}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 14700 }(8501, a) \) |
\(1\) | \(1\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) |
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sage:chi.jacobi_sum(n)
