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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(368, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,11,18]))
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pari:[g,chi] = znchar(Mod(149,368))
| Modulus: | \(368\) | |
| Conductor: | \(368\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(44\) |
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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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\(\chi_{368}(5,\cdot)\)
\(\chi_{368}(21,\cdot)\)
\(\chi_{368}(37,\cdot)\)
\(\chi_{368}(53,\cdot)\)
\(\chi_{368}(61,\cdot)\)
\(\chi_{368}(109,\cdot)\)
\(\chi_{368}(125,\cdot)\)
\(\chi_{368}(149,\cdot)\)
\(\chi_{368}(157,\cdot)\)
\(\chi_{368}(181,\cdot)\)
\(\chi_{368}(189,\cdot)\)
\(\chi_{368}(205,\cdot)\)
\(\chi_{368}(221,\cdot)\)
\(\chi_{368}(237,\cdot)\)
\(\chi_{368}(245,\cdot)\)
\(\chi_{368}(293,\cdot)\)
\(\chi_{368}(309,\cdot)\)
\(\chi_{368}(333,\cdot)\)
\(\chi_{368}(341,\cdot)\)
\(\chi_{368}(365,\cdot)\)
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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 ]
\((47,277,97)\) → \((1,i,e\left(\frac{9}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 368 }(149, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{25}{44}\right)\) |
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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)
