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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6897, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,27,44]))
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pari:[g,chi] = znchar(Mod(296,6897))
| Modulus: | \(6897\) | |
| Conductor: | \(6897\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(66\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
|
| Real: | no |
| Primitive: | yes |
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sage:chi.is_primitive()
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pari:#znconreyconductor(g,chi)==1
|
| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{6897}(197,\cdot)\)
\(\chi_{6897}(296,\cdot)\)
\(\chi_{6897}(824,\cdot)\)
\(\chi_{6897}(923,\cdot)\)
\(\chi_{6897}(1550,\cdot)\)
\(\chi_{6897}(2078,\cdot)\)
\(\chi_{6897}(2705,\cdot)\)
\(\chi_{6897}(2804,\cdot)\)
\(\chi_{6897}(3332,\cdot)\)
\(\chi_{6897}(3431,\cdot)\)
\(\chi_{6897}(3959,\cdot)\)
\(\chi_{6897}(4058,\cdot)\)
\(\chi_{6897}(4586,\cdot)\)
\(\chi_{6897}(4685,\cdot)\)
\(\chi_{6897}(5213,\cdot)\)
\(\chi_{6897}(5312,\cdot)\)
\(\chi_{6897}(5840,\cdot)\)
\(\chi_{6897}(5939,\cdot)\)
\(\chi_{6897}(6467,\cdot)\)
\(\chi_{6897}(6566,\cdot)\)
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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 ]
\((2300,970,3631)\) → \((-1,e\left(\frac{9}{22}\right),e\left(\frac{2}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 6897 }(296, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) |
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sage:chi.jacobi_sum(n)
