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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3724, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,31,14]))
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pari:[g,chi] = znchar(Mod(45,3724))
\(\chi_{3724}(45,\cdot)\)
\(\chi_{3724}(201,\cdot)\)
\(\chi_{3724}(577,\cdot)\)
\(\chi_{3724}(733,\cdot)\)
\(\chi_{3724}(1265,\cdot)\)
\(\chi_{3724}(1641,\cdot)\)
\(\chi_{3724}(1797,\cdot)\)
\(\chi_{3724}(2173,\cdot)\)
\(\chi_{3724}(2329,\cdot)\)
\(\chi_{3724}(2705,\cdot)\)
\(\chi_{3724}(3237,\cdot)\)
\(\chi_{3724}(3393,\cdot)\)
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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 ]
\((1863,3041,3137)\) → \((1,e\left(\frac{31}{42}\right),e\left(\frac{1}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 3724 }(45, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{3}{14}\right)\) |
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sage:chi.jacobi_sum(n)
