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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(287, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,7]))
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pari:[g,chi] = znchar(Mod(111,287))
| Modulus: | \(287\) | |
| Conductor: | \(287\) |
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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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\(\chi_{287}(6,\cdot)\)
\(\chi_{287}(13,\cdot)\)
\(\chi_{287}(34,\cdot)\)
\(\chi_{287}(48,\cdot)\)
\(\chi_{287}(69,\cdot)\)
\(\chi_{287}(76,\cdot)\)
\(\chi_{287}(97,\cdot)\)
\(\chi_{287}(104,\cdot)\)
\(\chi_{287}(111,\cdot)\)
\(\chi_{287}(153,\cdot)\)
\(\chi_{287}(181,\cdot)\)
\(\chi_{287}(188,\cdot)\)
\(\chi_{287}(216,\cdot)\)
\(\chi_{287}(258,\cdot)\)
\(\chi_{287}(265,\cdot)\)
\(\chi_{287}(272,\cdot)\)
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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 ]
\((206,211)\) → \((-1,e\left(\frac{7}{40}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 287 }(111, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(i\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) |
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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)
