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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([23,30]))
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pari:[g,chi] = znchar(Mod(219,695))
| Modulus: | \(695\) | |
| Conductor: | \(695\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| 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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{695}(34,\cdot)\)
\(\chi_{695}(44,\cdot)\)
\(\chi_{695}(64,\cdot)\)
\(\chi_{695}(79,\cdot)\)
\(\chi_{695}(129,\cdot)\)
\(\chi_{695}(184,\cdot)\)
\(\chi_{695}(194,\cdot)\)
\(\chi_{695}(204,\cdot)\)
\(\chi_{695}(219,\cdot)\)
\(\chi_{695}(239,\cdot)\)
\(\chi_{695}(264,\cdot)\)
\(\chi_{695}(284,\cdot)\)
\(\chi_{695}(314,\cdot)\)
\(\chi_{695}(369,\cdot)\)
\(\chi_{695}(384,\cdot)\)
\(\chi_{695}(394,\cdot)\)
\(\chi_{695}(409,\cdot)\)
\(\chi_{695}(469,\cdot)\)
\(\chi_{695}(474,\cdot)\)
\(\chi_{695}(494,\cdot)\)
\(\chi_{695}(529,\cdot)\)
\(\chi_{695}(619,\cdot)\)
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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 ]
\((557,141)\) → \((-1,e\left(\frac{15}{23}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 695 }(219, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{46}\right)\) | \(e\left(\frac{11}{46}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{5}{46}\right)\) | \(e\left(\frac{21}{46}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{25}{46}\right)\) | \(e\left(\frac{11}{46}\right)\) |
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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)
