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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([0,11,24]))
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pari:[g,chi] = znchar(Mod(133,368))
| 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)
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| 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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\(\chi_{368}(13,\cdot)\)
\(\chi_{368}(29,\cdot)\)
\(\chi_{368}(77,\cdot)\)
\(\chi_{368}(85,\cdot)\)
\(\chi_{368}(101,\cdot)\)
\(\chi_{368}(117,\cdot)\)
\(\chi_{368}(133,\cdot)\)
\(\chi_{368}(141,\cdot)\)
\(\chi_{368}(165,\cdot)\)
\(\chi_{368}(173,\cdot)\)
\(\chi_{368}(197,\cdot)\)
\(\chi_{368}(213,\cdot)\)
\(\chi_{368}(261,\cdot)\)
\(\chi_{368}(269,\cdot)\)
\(\chi_{368}(285,\cdot)\)
\(\chi_{368}(301,\cdot)\)
\(\chi_{368}(317,\cdot)\)
\(\chi_{368}(325,\cdot)\)
\(\chi_{368}(349,\cdot)\)
\(\chi_{368}(357,\cdot)\)
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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 ]
\((47,277,97)\) → \((1,i,e\left(\frac{6}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 368 }(133, a) \) |
\(1\) | \(1\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{15}{44}\right)\) |
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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)
