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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([0,33,44,12]))
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pari:[g,chi] = znchar(Mod(4,2415))
\(\chi_{2415}(4,\cdot)\)
\(\chi_{2415}(289,\cdot)\)
\(\chi_{2415}(394,\cdot)\)
\(\chi_{2415}(499,\cdot)\)
\(\chi_{2415}(604,\cdot)\)
\(\chi_{2415}(634,\cdot)\)
\(\chi_{2415}(739,\cdot)\)
\(\chi_{2415}(814,\cdot)\)
\(\chi_{2415}(844,\cdot)\)
\(\chi_{2415}(949,\cdot)\)
\(\chi_{2415}(1024,\cdot)\)
\(\chi_{2415}(1129,\cdot)\)
\(\chi_{2415}(1159,\cdot)\)
\(\chi_{2415}(1369,\cdot)\)
\(\chi_{2415}(1444,\cdot)\)
\(\chi_{2415}(1474,\cdot)\)
\(\chi_{2415}(1549,\cdot)\)
\(\chi_{2415}(1789,\cdot)\)
\(\chi_{2415}(1894,\cdot)\)
\(\chi_{2415}(2074,\cdot)\)
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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 ]
\((806,967,346,1891)\) → \((1,-1,e\left(\frac{2}{3}\right),e\left(\frac{2}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(26\) |
| \( \chi_{ 2415 }(4, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(-1\) | \(e\left(\frac{8}{33}\right)\) |
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sage:chi.jacobi_sum(n)
