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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2415, base_ring=CyclotomicField(132))
M = H._module
chi = DirichletCharacter(H, M([0,99,44,12]))
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pari:[g,chi] = znchar(Mod(163,2415))
\(\chi_{2415}(58,\cdot)\)
\(\chi_{2415}(142,\cdot)\)
\(\chi_{2415}(163,\cdot)\)
\(\chi_{2415}(193,\cdot)\)
\(\chi_{2415}(403,\cdot)\)
\(\chi_{2415}(478,\cdot)\)
\(\chi_{2415}(487,\cdot)\)
\(\chi_{2415}(508,\cdot)\)
\(\chi_{2415}(583,\cdot)\)
\(\chi_{2415}(772,\cdot)\)
\(\chi_{2415}(823,\cdot)\)
\(\chi_{2415}(877,\cdot)\)
\(\chi_{2415}(928,\cdot)\)
\(\chi_{2415}(982,\cdot)\)
\(\chi_{2415}(1087,\cdot)\)
\(\chi_{2415}(1108,\cdot)\)
\(\chi_{2415}(1117,\cdot)\)
\(\chi_{2415}(1222,\cdot)\)
\(\chi_{2415}(1297,\cdot)\)
\(\chi_{2415}(1327,\cdot)\)
\(\chi_{2415}(1432,\cdot)\)
\(\chi_{2415}(1453,\cdot)\)
\(\chi_{2415}(1507,\cdot)\)
\(\chi_{2415}(1612,\cdot)\)
\(\chi_{2415}(1642,\cdot)\)
\(\chi_{2415}(1738,\cdot)\)
\(\chi_{2415}(1843,\cdot)\)
\(\chi_{2415}(1852,\cdot)\)
\(\chi_{2415}(1927,\cdot)\)
\(\chi_{2415}(1948,\cdot)\)
...
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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 ]
\((806,967,346,1891)\) → \((1,-i,e\left(\frac{1}{3}\right),e\left(\frac{1}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(26\) |
| \( \chi_{ 2415 }(163, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{79}{132}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{95}{132}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(-i\) | \(e\left(\frac{4}{33}\right)\) |
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sage:chi.jacobi_sum(n)
