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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(18225, base_ring=CyclotomicField(486))
M = H._module
chi = DirichletCharacter(H, M([148,243]))
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pari:[g,chi] = znchar(Mod(2824,18225))
\(\chi_{18225}(49,\cdot)\)
\(\chi_{18225}(124,\cdot)\)
\(\chi_{18225}(274,\cdot)\)
\(\chi_{18225}(349,\cdot)\)
\(\chi_{18225}(499,\cdot)\)
\(\chi_{18225}(574,\cdot)\)
\(\chi_{18225}(724,\cdot)\)
\(\chi_{18225}(799,\cdot)\)
\(\chi_{18225}(949,\cdot)\)
\(\chi_{18225}(1024,\cdot)\)
\(\chi_{18225}(1174,\cdot)\)
\(\chi_{18225}(1249,\cdot)\)
\(\chi_{18225}(1399,\cdot)\)
\(\chi_{18225}(1474,\cdot)\)
\(\chi_{18225}(1624,\cdot)\)
\(\chi_{18225}(1699,\cdot)\)
\(\chi_{18225}(1849,\cdot)\)
\(\chi_{18225}(1924,\cdot)\)
\(\chi_{18225}(2074,\cdot)\)
\(\chi_{18225}(2149,\cdot)\)
\(\chi_{18225}(2299,\cdot)\)
\(\chi_{18225}(2374,\cdot)\)
\(\chi_{18225}(2524,\cdot)\)
\(\chi_{18225}(2599,\cdot)\)
\(\chi_{18225}(2749,\cdot)\)
\(\chi_{18225}(2824,\cdot)\)
\(\chi_{18225}(2974,\cdot)\)
\(\chi_{18225}(3049,\cdot)\)
\(\chi_{18225}(3199,\cdot)\)
\(\chi_{18225}(3274,\cdot)\)
...
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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 ]
\((4376,13852)\) → \((e\left(\frac{74}{243}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 18225 }(2824, a) \) |
\(1\) | \(1\) | \(e\left(\frac{391}{486}\right)\) | \(e\left(\frac{148}{243}\right)\) | \(e\left(\frac{235}{486}\right)\) | \(e\left(\frac{67}{162}\right)\) | \(e\left(\frac{44}{243}\right)\) | \(e\left(\frac{293}{486}\right)\) | \(e\left(\frac{70}{243}\right)\) | \(e\left(\frac{53}{243}\right)\) | \(e\left(\frac{89}{162}\right)\) | \(e\left(\frac{68}{81}\right)\) |
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sage:chi.jacobi_sum(n)
