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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2415, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,33,0,12]))
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pari:[g,chi] = znchar(Mod(8,2415))
\(\chi_{2415}(8,\cdot)\)
\(\chi_{2415}(197,\cdot)\)
\(\chi_{2415}(302,\cdot)\)
\(\chi_{2415}(407,\cdot)\)
\(\chi_{2415}(512,\cdot)\)
\(\chi_{2415}(533,\cdot)\)
\(\chi_{2415}(722,\cdot)\)
\(\chi_{2415}(932,\cdot)\)
\(\chi_{2415}(1037,\cdot)\)
\(\chi_{2415}(1163,\cdot)\)
\(\chi_{2415}(1268,\cdot)\)
\(\chi_{2415}(1352,\cdot)\)
\(\chi_{2415}(1373,\cdot)\)
\(\chi_{2415}(1457,\cdot)\)
\(\chi_{2415}(1478,\cdot)\)
\(\chi_{2415}(1688,\cdot)\)
\(\chi_{2415}(1898,\cdot)\)
\(\chi_{2415}(1982,\cdot)\)
\(\chi_{2415}(2003,\cdot)\)
\(\chi_{2415}(2318,\cdot)\)
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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 ]
\((806,967,346,1891)\) → \((-1,-i,1,e\left(\frac{3}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(26\) |
| \( \chi_{ 2415 }(8, a) \) |
\(1\) | \(1\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(-i\) | \(e\left(\frac{19}{22}\right)\) |
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sage:chi.jacobi_sum(n)
