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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5200, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,15,27,20]))
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pari:[g,chi] = znchar(Mod(2187,5200))
| Modulus: | \(5200\) | |
| Conductor: | \(5200\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(60\) |
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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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\(\chi_{5200}(3,\cdot)\)
\(\chi_{5200}(347,\cdot)\)
\(\chi_{5200}(1147,\cdot)\)
\(\chi_{5200}(1283,\cdot)\)
\(\chi_{5200}(1387,\cdot)\)
\(\chi_{5200}(2083,\cdot)\)
\(\chi_{5200}(2187,\cdot)\)
\(\chi_{5200}(2323,\cdot)\)
\(\chi_{5200}(2427,\cdot)\)
\(\chi_{5200}(3123,\cdot)\)
\(\chi_{5200}(3227,\cdot)\)
\(\chi_{5200}(3363,\cdot)\)
\(\chi_{5200}(3467,\cdot)\)
\(\chi_{5200}(4163,\cdot)\)
\(\chi_{5200}(4267,\cdot)\)
\(\chi_{5200}(4403,\cdot)\)
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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 ]
\((1951,1301,4577,1601)\) → \((-1,i,e\left(\frac{9}{20}\right),e\left(\frac{1}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 5200 }(2187, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{59}{60}\right)\) |
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sage:chi.jacobi_sum(n)
