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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1288, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,33,11,36]))
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pari:[g,chi] = znchar(Mod(731,1288))
| Modulus: | \(1288\) | |
| Conductor: | \(1288\) |
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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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\(\chi_{1288}(3,\cdot)\)
\(\chi_{1288}(59,\cdot)\)
\(\chi_{1288}(75,\cdot)\)
\(\chi_{1288}(131,\cdot)\)
\(\chi_{1288}(187,\cdot)\)
\(\chi_{1288}(243,\cdot)\)
\(\chi_{1288}(395,\cdot)\)
\(\chi_{1288}(579,\cdot)\)
\(\chi_{1288}(675,\cdot)\)
\(\chi_{1288}(731,\cdot)\)
\(\chi_{1288}(859,\cdot)\)
\(\chi_{1288}(899,\cdot)\)
\(\chi_{1288}(915,\cdot)\)
\(\chi_{1288}(955,\cdot)\)
\(\chi_{1288}(1067,\cdot)\)
\(\chi_{1288}(1083,\cdot)\)
\(\chi_{1288}(1139,\cdot)\)
\(\chi_{1288}(1179,\cdot)\)
\(\chi_{1288}(1235,\cdot)\)
\(\chi_{1288}(1251,\cdot)\)
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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 ]
\((967,645,185,281)\) → \((-1,-1,e\left(\frac{1}{6}\right),e\left(\frac{6}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(25\) | \(27\) |
| \( \chi_{ 1288 }(731, a) \) |
\(1\) | \(1\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{15}{22}\right)\) |
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sage:chi.jacobi_sum(n)
