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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(695, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([23,37]))
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pari:[g,chi] = znchar(Mod(689,695))
| Modulus: | \(695\) | |
| Conductor: | \(695\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(46\) |
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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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\(\chi_{695}(14,\cdot)\)
\(\chi_{695}(39,\cdot)\)
\(\chi_{695}(59,\cdot)\)
\(\chi_{695}(74,\cdot)\)
\(\chi_{695}(84,\cdot)\)
\(\chi_{695}(94,\cdot)\)
\(\chi_{695}(149,\cdot)\)
\(\chi_{695}(199,\cdot)\)
\(\chi_{695}(214,\cdot)\)
\(\chi_{695}(234,\cdot)\)
\(\chi_{695}(244,\cdot)\)
\(\chi_{695}(354,\cdot)\)
\(\chi_{695}(444,\cdot)\)
\(\chi_{695}(479,\cdot)\)
\(\chi_{695}(499,\cdot)\)
\(\chi_{695}(504,\cdot)\)
\(\chi_{695}(564,\cdot)\)
\(\chi_{695}(579,\cdot)\)
\(\chi_{695}(589,\cdot)\)
\(\chi_{695}(604,\cdot)\)
\(\chi_{695}(659,\cdot)\)
\(\chi_{695}(689,\cdot)\)
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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 ]
\((557,141)\) → \((-1,e\left(\frac{37}{46}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 695 }(689, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{33}{46}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{45}{46}\right)\) |
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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)
