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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(212, base_ring=CyclotomicField(52))
M = H._module
chi = DirichletCharacter(H, M([26,31]))
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pari:[g,chi] = znchar(Mod(127,212))
| Modulus: | \(212\) | |
| Conductor: | \(212\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(52\) |
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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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\(\chi_{212}(3,\cdot)\)
\(\chi_{212}(19,\cdot)\)
\(\chi_{212}(27,\cdot)\)
\(\chi_{212}(31,\cdot)\)
\(\chi_{212}(35,\cdot)\)
\(\chi_{212}(39,\cdot)\)
\(\chi_{212}(51,\cdot)\)
\(\chi_{212}(55,\cdot)\)
\(\chi_{212}(67,\cdot)\)
\(\chi_{212}(71,\cdot)\)
\(\chi_{212}(75,\cdot)\)
\(\chi_{212}(79,\cdot)\)
\(\chi_{212}(87,\cdot)\)
\(\chi_{212}(103,\cdot)\)
\(\chi_{212}(111,\cdot)\)
\(\chi_{212}(127,\cdot)\)
\(\chi_{212}(139,\cdot)\)
\(\chi_{212}(147,\cdot)\)
\(\chi_{212}(151,\cdot)\)
\(\chi_{212}(167,\cdot)\)
\(\chi_{212}(171,\cdot)\)
\(\chi_{212}(179,\cdot)\)
\(\chi_{212}(191,\cdot)\)
\(\chi_{212}(207,\cdot)\)
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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 ]
\((107,161)\) → \((-1,e\left(\frac{31}{52}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 212 }(127, a) \) |
\(1\) | \(1\) | \(e\left(\frac{33}{52}\right)\) | \(e\left(\frac{1}{52}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{29}{52}\right)\) | \(e\left(\frac{25}{52}\right)\) |
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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)
