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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3864, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,33,33,55,51]))
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pari:[g,chi] = znchar(Mod(2315,3864))
| Modulus: | \(3864\) | |
| Conductor: | \(3864\) |
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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
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| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{3864}(227,\cdot)\)
\(\chi_{3864}(467,\cdot)\)
\(\chi_{3864}(563,\cdot)\)
\(\chi_{3864}(635,\cdot)\)
\(\chi_{3864}(803,\cdot)\)
\(\chi_{3864}(971,\cdot)\)
\(\chi_{3864}(1307,\cdot)\)
\(\chi_{3864}(1571,\cdot)\)
\(\chi_{3864}(1643,\cdot)\)
\(\chi_{3864}(1739,\cdot)\)
\(\chi_{3864}(1811,\cdot)\)
\(\chi_{3864}(1907,\cdot)\)
\(\chi_{3864}(2075,\cdot)\)
\(\chi_{3864}(2315,\cdot)\)
\(\chi_{3864}(2411,\cdot)\)
\(\chi_{3864}(2747,\cdot)\)
\(\chi_{3864}(2915,\cdot)\)
\(\chi_{3864}(2987,\cdot)\)
\(\chi_{3864}(3323,\cdot)\)
\(\chi_{3864}(3419,\cdot)\)
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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 ]
\((967,1933,1289,2761,2857)\) → \((-1,-1,-1,e\left(\frac{5}{6}\right),e\left(\frac{17}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3864 }(2315, a) \) |
\(1\) | \(1\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{3}{11}\right)\) |
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sage:chi.jacobi_sum(n)
