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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7448, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,0,31,7]))
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pari:[g,chi] = znchar(Mod(6905,7448))
\(\chi_{7448}(145,\cdot)\)
\(\chi_{7448}(1209,\cdot)\)
\(\chi_{7448}(1585,\cdot)\)
\(\chi_{7448}(2649,\cdot)\)
\(\chi_{7448}(3337,\cdot)\)
\(\chi_{7448}(3713,\cdot)\)
\(\chi_{7448}(4401,\cdot)\)
\(\chi_{7448}(4777,\cdot)\)
\(\chi_{7448}(5465,\cdot)\)
\(\chi_{7448}(5841,\cdot)\)
\(\chi_{7448}(6529,\cdot)\)
\(\chi_{7448}(6905,\cdot)\)
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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 ]
\((1863,3725,3041,3137)\) → \((1,1,e\left(\frac{31}{42}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 7448 }(6905, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) |
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sage:chi.jacobi_sum(n)
