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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([33,33,33,22,54]))
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pari:[g,chi] = znchar(Mod(443,3864))
| 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
|
| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{3864}(179,\cdot)\)
\(\chi_{3864}(347,\cdot)\)
\(\chi_{3864}(443,\cdot)\)
\(\chi_{3864}(515,\cdot)\)
\(\chi_{3864}(611,\cdot)\)
\(\chi_{3864}(683,\cdot)\)
\(\chi_{3864}(947,\cdot)\)
\(\chi_{3864}(1283,\cdot)\)
\(\chi_{3864}(1451,\cdot)\)
\(\chi_{3864}(1619,\cdot)\)
\(\chi_{3864}(1691,\cdot)\)
\(\chi_{3864}(1787,\cdot)\)
\(\chi_{3864}(2027,\cdot)\)
\(\chi_{3864}(2699,\cdot)\)
\(\chi_{3864}(2795,\cdot)\)
\(\chi_{3864}(3131,\cdot)\)
\(\chi_{3864}(3203,\cdot)\)
\(\chi_{3864}(3371,\cdot)\)
\(\chi_{3864}(3707,\cdot)\)
\(\chi_{3864}(3803,\cdot)\)
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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 ]
\((967,1933,1289,2761,2857)\) → \((-1,-1,-1,e\left(\frac{1}{3}\right),e\left(\frac{9}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3864 }(443, a) \) |
\(1\) | \(1\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{7}{22}\right)\) |
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sage:chi.jacobi_sum(n)
