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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2720, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,6,12,1]))
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pari:[g,chi] = znchar(Mod(3,2720))
| Modulus: | \(2720\) | |
| Conductor: | \(2720\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(16\) |
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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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{2720}(3,\cdot)\)
\(\chi_{2720}(27,\cdot)\)
\(\chi_{2720}(243,\cdot)\)
\(\chi_{2720}(347,\cdot)\)
\(\chi_{2720}(403,\cdot)\)
\(\chi_{2720}(643,\cdot)\)
\(\chi_{2720}(907,\cdot)\)
\(\chi_{2720}(2187,\cdot)\)
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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 ]
\((511,1701,2177,1601)\) → \((-1,e\left(\frac{3}{8}\right),-i,e\left(\frac{1}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 2720 }(3, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(-1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) |
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sage:chi.jacobi_sum(n)
