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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(30345, base_ring=CyclotomicField(408))
M = H._module
chi = DirichletCharacter(H, M([0,102,68,267]))
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pari:[g,chi] = znchar(Mod(1522,30345))
\(\chi_{30345}(502,\cdot)\)
\(\chi_{30345}(892,\cdot)\)
\(\chi_{30345}(1018,\cdot)\)
\(\chi_{30345}(1522,\cdot)\)
\(\chi_{30345}(1657,\cdot)\)
\(\chi_{30345}(1753,\cdot)\)
\(\chi_{30345}(1783,\cdot)\)
\(\chi_{30345}(2287,\cdot)\)
\(\chi_{30345}(2518,\cdot)\)
\(\chi_{30345}(2677,\cdot)\)
\(\chi_{30345}(2803,\cdot)\)
\(\chi_{30345}(3307,\cdot)\)
\(\chi_{30345}(3442,\cdot)\)
\(\chi_{30345}(3538,\cdot)\)
\(\chi_{30345}(3568,\cdot)\)
\(\chi_{30345}(4072,\cdot)\)
\(\chi_{30345}(4303,\cdot)\)
\(\chi_{30345}(4462,\cdot)\)
\(\chi_{30345}(4588,\cdot)\)
\(\chi_{30345}(5227,\cdot)\)
\(\chi_{30345}(5323,\cdot)\)
\(\chi_{30345}(5353,\cdot)\)
\(\chi_{30345}(5857,\cdot)\)
\(\chi_{30345}(6088,\cdot)\)
\(\chi_{30345}(6247,\cdot)\)
\(\chi_{30345}(6373,\cdot)\)
\(\chi_{30345}(6877,\cdot)\)
\(\chi_{30345}(7012,\cdot)\)
\(\chi_{30345}(7108,\cdot)\)
\(\chi_{30345}(7138,\cdot)\)
...
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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 ]
\((20231,24277,4336,28036)\) → \((1,i,e\left(\frac{1}{6}\right),e\left(\frac{89}{136}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(19\) | \(22\) | \(23\) | \(26\) |
| \( \chi_{ 30345 }(1522, a) \) |
\(1\) | \(1\) | \(e\left(\frac{47}{51}\right)\) | \(e\left(\frac{43}{51}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{293}{408}\right)\) | \(e\left(\frac{35}{68}\right)\) | \(e\left(\frac{35}{51}\right)\) | \(e\left(\frac{101}{204}\right)\) | \(e\left(\frac{87}{136}\right)\) | \(e\left(\frac{7}{408}\right)\) | \(e\left(\frac{89}{204}\right)\) |
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sage:chi.jacobi_sum(n)
