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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(325, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([48,25]))
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pari:[g,chi] = znchar(Mod(136,325))
| Modulus: | \(325\) | |
| Conductor: | \(325\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(60\) |
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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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{325}(6,\cdot)\)
\(\chi_{325}(11,\cdot)\)
\(\chi_{325}(41,\cdot)\)
\(\chi_{325}(46,\cdot)\)
\(\chi_{325}(71,\cdot)\)
\(\chi_{325}(106,\cdot)\)
\(\chi_{325}(111,\cdot)\)
\(\chi_{325}(136,\cdot)\)
\(\chi_{325}(141,\cdot)\)
\(\chi_{325}(171,\cdot)\)
\(\chi_{325}(206,\cdot)\)
\(\chi_{325}(236,\cdot)\)
\(\chi_{325}(241,\cdot)\)
\(\chi_{325}(266,\cdot)\)
\(\chi_{325}(271,\cdot)\)
\(\chi_{325}(306,\cdot)\)
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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 ]
\((27,301)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{5}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 325 }(136, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) |
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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)
