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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(451, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([12,39]))
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pari:[g,chi] = znchar(Mod(294,451))
| Modulus: | \(451\) | |
| Conductor: | \(451\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(40\) |
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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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\(\chi_{451}(19,\cdot)\)
\(\chi_{451}(24,\cdot)\)
\(\chi_{451}(63,\cdot)\)
\(\chi_{451}(94,\cdot)\)
\(\chi_{451}(95,\cdot)\)
\(\chi_{451}(112,\cdot)\)
\(\chi_{451}(117,\cdot)\)
\(\chi_{451}(134,\cdot)\)
\(\chi_{451}(138,\cdot)\)
\(\chi_{451}(149,\cdot)\)
\(\chi_{451}(193,\cdot)\)
\(\chi_{451}(222,\cdot)\)
\(\chi_{451}(239,\cdot)\)
\(\chi_{451}(293,\cdot)\)
\(\chi_{451}(294,\cdot)\)
\(\chi_{451}(315,\cdot)\)
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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 ]
\((288,375)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{39}{40}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 451 }(294, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{13}{40}\right)\) |
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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)
