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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(448, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,15,32]))
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pari:[g,chi] = znchar(Mod(53,448))
| Modulus: | \(448\) | |
| Conductor: | \(448\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(48\) |
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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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\(\chi_{448}(37,\cdot)\)
\(\chi_{448}(53,\cdot)\)
\(\chi_{448}(93,\cdot)\)
\(\chi_{448}(109,\cdot)\)
\(\chi_{448}(149,\cdot)\)
\(\chi_{448}(165,\cdot)\)
\(\chi_{448}(205,\cdot)\)
\(\chi_{448}(221,\cdot)\)
\(\chi_{448}(261,\cdot)\)
\(\chi_{448}(277,\cdot)\)
\(\chi_{448}(317,\cdot)\)
\(\chi_{448}(333,\cdot)\)
\(\chi_{448}(373,\cdot)\)
\(\chi_{448}(389,\cdot)\)
\(\chi_{448}(429,\cdot)\)
\(\chi_{448}(445,\cdot)\)
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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 ]
\((127,197,129)\) → \((1,e\left(\frac{5}{16}\right),e\left(\frac{2}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 448 }(53, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(i\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{7}{24}\right)\) |
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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)
