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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(417, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([23,36]))
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pari:[g,chi] = znchar(Mod(173,417))
| Modulus: | \(417\) | |
| Conductor: | \(417\) |
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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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\(\chi_{417}(44,\cdot)\)
\(\chi_{417}(65,\cdot)\)
\(\chi_{417}(77,\cdot)\)
\(\chi_{417}(80,\cdot)\)
\(\chi_{417}(116,\cdot)\)
\(\chi_{417}(125,\cdot)\)
\(\chi_{417}(131,\cdot)\)
\(\chi_{417}(173,\cdot)\)
\(\chi_{417}(191,\cdot)\)
\(\chi_{417}(194,\cdot)\)
\(\chi_{417}(203,\cdot)\)
\(\chi_{417}(218,\cdot)\)
\(\chi_{417}(230,\cdot)\)
\(\chi_{417}(239,\cdot)\)
\(\chi_{417}(245,\cdot)\)
\(\chi_{417}(251,\cdot)\)
\(\chi_{417}(284,\cdot)\)
\(\chi_{417}(314,\cdot)\)
\(\chi_{417}(323,\cdot)\)
\(\chi_{417}(335,\cdot)\)
\(\chi_{417}(341,\cdot)\)
\(\chi_{417}(407,\cdot)\)
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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 ]
\((140,280)\) → \((-1,e\left(\frac{18}{23}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 417 }(173, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{13}{46}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{37}{46}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{39}{46}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{45}{46}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{19}{46}\right)\) | \(e\left(\frac{3}{23}\right)\) |
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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)
