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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(312, base_ring=CyclotomicField(2))
M = H._module
chi = DirichletCharacter(H, M([0,1,1,1]))
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pari:[g,chi] = znchar(Mod(77,312))
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sage:kronecker_character(-312)
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pari:znchartokronecker(g,chi)
\(\displaystyle\left(\frac{-312}{\bullet}\right)\)
| Modulus: | \(312\) | |
| Conductor: | \(312\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(2\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
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| Real: | yes |
| 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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\(\chi_{312}(77,\cdot)\)
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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 ]
\((79,157,209,145)\) → \((1,-1,-1,-1)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 312 }(77, a) \) |
\(-1\) | \(1\) | \(-1\) | \(-1\) | \(-1\) | \(-1\) | \(1\) | \(-1\) | \(1\) | \(1\) | \(-1\) | \(1\) |
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sage:chi.jacobi_sum(n)
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sage:chi.gauss_sum(a)
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pari:znchargauss(g,chi,a)
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sage:chi.jacobi_sum(n)
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sage:chi.kloosterman_sum(a,b)
