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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2415, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,0,11,42]))
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pari:[g,chi] = znchar(Mod(1991,2415))
\(\chi_{2415}(26,\cdot)\)
\(\chi_{2415}(101,\cdot)\)
\(\chi_{2415}(131,\cdot)\)
\(\chi_{2415}(236,\cdot)\)
\(\chi_{2415}(311,\cdot)\)
\(\chi_{2415}(416,\cdot)\)
\(\chi_{2415}(446,\cdot)\)
\(\chi_{2415}(656,\cdot)\)
\(\chi_{2415}(731,\cdot)\)
\(\chi_{2415}(761,\cdot)\)
\(\chi_{2415}(836,\cdot)\)
\(\chi_{2415}(1076,\cdot)\)
\(\chi_{2415}(1181,\cdot)\)
\(\chi_{2415}(1361,\cdot)\)
\(\chi_{2415}(1706,\cdot)\)
\(\chi_{2415}(1991,\cdot)\)
\(\chi_{2415}(2096,\cdot)\)
\(\chi_{2415}(2201,\cdot)\)
\(\chi_{2415}(2306,\cdot)\)
\(\chi_{2415}(2336,\cdot)\)
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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 ]
\((806,967,346,1891)\) → \((-1,1,e\left(\frac{1}{6}\right),e\left(\frac{7}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(26\) |
| \( \chi_{ 2415 }(1991, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(1\) | \(e\left(\frac{17}{33}\right)\) |
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sage:chi.jacobi_sum(n)
