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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(663, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,40,15]))
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pari:[g,chi] = znchar(Mod(413,663))
| Modulus: | \(663\) | |
| Conductor: | \(663\) |
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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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\(\chi_{663}(23,\cdot)\)
\(\chi_{663}(56,\cdot)\)
\(\chi_{663}(62,\cdot)\)
\(\chi_{663}(95,\cdot)\)
\(\chi_{663}(173,\cdot)\)
\(\chi_{663}(218,\cdot)\)
\(\chi_{663}(296,\cdot)\)
\(\chi_{663}(329,\cdot)\)
\(\chi_{663}(335,\cdot)\)
\(\chi_{663}(368,\cdot)\)
\(\chi_{663}(413,\cdot)\)
\(\chi_{663}(452,\cdot)\)
\(\chi_{663}(524,\cdot)\)
\(\chi_{663}(530,\cdot)\)
\(\chi_{663}(602,\cdot)\)
\(\chi_{663}(641,\cdot)\)
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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 ]
\((443,613,547)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{5}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(19\) |
| \( \chi_{ 663 }(413, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{24}\right)\) |
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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)
