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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(896, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,23,16]))
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pari:[g,chi] = znchar(Mod(531,896))
| Modulus: | \(896\) | |
| Conductor: | \(896\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(32\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
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| 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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\(\chi_{896}(27,\cdot)\)
\(\chi_{896}(83,\cdot)\)
\(\chi_{896}(139,\cdot)\)
\(\chi_{896}(195,\cdot)\)
\(\chi_{896}(251,\cdot)\)
\(\chi_{896}(307,\cdot)\)
\(\chi_{896}(363,\cdot)\)
\(\chi_{896}(419,\cdot)\)
\(\chi_{896}(475,\cdot)\)
\(\chi_{896}(531,\cdot)\)
\(\chi_{896}(587,\cdot)\)
\(\chi_{896}(643,\cdot)\)
\(\chi_{896}(699,\cdot)\)
\(\chi_{896}(755,\cdot)\)
\(\chi_{896}(811,\cdot)\)
\(\chi_{896}(867,\cdot)\)
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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 ]
\((127,645,129)\) → \((-1,e\left(\frac{23}{32}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 896 }(531, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) |
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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)
