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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5625, base_ring=CyclotomicField(150))
M = H._module
chi = DirichletCharacter(H, M([50,141]))
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pari:[g,chi] = znchar(Mod(49,5625))
\(\chi_{5625}(49,\cdot)\)
\(\chi_{5625}(274,\cdot)\)
\(\chi_{5625}(349,\cdot)\)
\(\chi_{5625}(574,\cdot)\)
\(\chi_{5625}(724,\cdot)\)
\(\chi_{5625}(799,\cdot)\)
\(\chi_{5625}(949,\cdot)\)
\(\chi_{5625}(1024,\cdot)\)
\(\chi_{5625}(1174,\cdot)\)
\(\chi_{5625}(1399,\cdot)\)
\(\chi_{5625}(1474,\cdot)\)
\(\chi_{5625}(1699,\cdot)\)
\(\chi_{5625}(1849,\cdot)\)
\(\chi_{5625}(1924,\cdot)\)
\(\chi_{5625}(2074,\cdot)\)
\(\chi_{5625}(2149,\cdot)\)
\(\chi_{5625}(2299,\cdot)\)
\(\chi_{5625}(2524,\cdot)\)
\(\chi_{5625}(2599,\cdot)\)
\(\chi_{5625}(2824,\cdot)\)
\(\chi_{5625}(2974,\cdot)\)
\(\chi_{5625}(3049,\cdot)\)
\(\chi_{5625}(3199,\cdot)\)
\(\chi_{5625}(3274,\cdot)\)
\(\chi_{5625}(3424,\cdot)\)
\(\chi_{5625}(3649,\cdot)\)
\(\chi_{5625}(3724,\cdot)\)
\(\chi_{5625}(3949,\cdot)\)
\(\chi_{5625}(4099,\cdot)\)
\(\chi_{5625}(4174,\cdot)\)
...
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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 ]
\((4376,1252)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{47}{50}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 5625 }(49, a) \) |
\(1\) | \(1\) | \(e\left(\frac{41}{150}\right)\) | \(e\left(\frac{41}{75}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{58}{75}\right)\) | \(e\left(\frac{49}{150}\right)\) | \(e\left(\frac{38}{75}\right)\) | \(e\left(\frac{7}{75}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{23}{25}\right)\) |
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sage:chi.jacobi_sum(n)
