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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2442, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,42,35]))
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pari:[g,chi] = znchar(Mod(1657,2442))
\(\chi_{2442}(193,\cdot)\)
\(\chi_{2442}(325,\cdot)\)
\(\chi_{2442}(415,\cdot)\)
\(\chi_{2442}(541,\cdot)\)
\(\chi_{2442}(547,\cdot)\)
\(\chi_{2442}(865,\cdot)\)
\(\chi_{2442}(985,\cdot)\)
\(\chi_{2442}(1207,\cdot)\)
\(\chi_{2442}(1525,\cdot)\)
\(\chi_{2442}(1531,\cdot)\)
\(\chi_{2442}(1657,\cdot)\)
\(\chi_{2442}(1975,\cdot)\)
\(\chi_{2442}(2191,\cdot)\)
\(\chi_{2442}(2197,\cdot)\)
\(\chi_{2442}(2317,\cdot)\)
\(\chi_{2442}(2323,\cdot)\)
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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 ]
\((815,1333,1519)\) → \((1,e\left(\frac{7}{10}\right),e\left(\frac{7}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 2442 }(1657, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(-i\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{47}{60}\right)\) |
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sage:chi.jacobi_sum(n)
