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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(820, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,5,13]))
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pari:[g,chi] = znchar(Mod(207,820))
| Modulus: | \(820\) | |
| Conductor: | \(820\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(20\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
|
| 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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\(\chi_{820}(103,\cdot)\)
\(\chi_{820}(203,\cdot)\)
\(\chi_{820}(207,\cdot)\)
\(\chi_{820}(307,\cdot)\)
\(\chi_{820}(487,\cdot)\)
\(\chi_{820}(607,\cdot)\)
\(\chi_{820}(623,\cdot)\)
\(\chi_{820}(743,\cdot)\)
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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 ]
\((411,657,621)\) → \((-1,i,e\left(\frac{13}{20}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 820 }(207, a) \) |
\(1\) | \(1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(1\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(1\) |
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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)
