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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(925, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([54,55]))
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pari:[g,chi] = znchar(Mod(569,925))
| Modulus: | \(925\) | |
| Conductor: | \(925\) |
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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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\(\chi_{925}(14,\cdot)\)
\(\chi_{925}(29,\cdot)\)
\(\chi_{925}(119,\cdot)\)
\(\chi_{925}(134,\cdot)\)
\(\chi_{925}(214,\cdot)\)
\(\chi_{925}(304,\cdot)\)
\(\chi_{925}(319,\cdot)\)
\(\chi_{925}(384,\cdot)\)
\(\chi_{925}(489,\cdot)\)
\(\chi_{925}(504,\cdot)\)
\(\chi_{925}(569,\cdot)\)
\(\chi_{925}(584,\cdot)\)
\(\chi_{925}(689,\cdot)\)
\(\chi_{925}(754,\cdot)\)
\(\chi_{925}(769,\cdot)\)
\(\chi_{925}(859,\cdot)\)
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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 ]
\((852,76)\) → \((e\left(\frac{9}{10}\right),e\left(\frac{11}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 925 }(569, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{11}{60}\right)\) |
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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)
