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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(473, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,37]))
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pari:[g,chi] = znchar(Mod(450,473))
| Modulus: | \(473\) | |
| Conductor: | \(473\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(42\) |
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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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\(\chi_{473}(76,\cdot)\)
\(\chi_{473}(98,\cdot)\)
\(\chi_{473}(120,\cdot)\)
\(\chi_{473}(175,\cdot)\)
\(\chi_{473}(241,\cdot)\)
\(\chi_{473}(263,\cdot)\)
\(\chi_{473}(329,\cdot)\)
\(\chi_{473}(362,\cdot)\)
\(\chi_{473}(373,\cdot)\)
\(\chi_{473}(406,\cdot)\)
\(\chi_{473}(417,\cdot)\)
\(\chi_{473}(450,\cdot)\)
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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 ]
\((431,89)\) → \((-1,e\left(\frac{37}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 473 }(450, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{19}{42}\right)\) |
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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)
