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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(507, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([78,155]))
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pari:[g,chi] = znchar(Mod(254,507))
| Modulus: | \(507\) | |
| Conductor: | \(507\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(156\) |
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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
|
| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
|
\(\chi_{507}(2,\cdot)\)
\(\chi_{507}(11,\cdot)\)
\(\chi_{507}(20,\cdot)\)
\(\chi_{507}(32,\cdot)\)
\(\chi_{507}(41,\cdot)\)
\(\chi_{507}(50,\cdot)\)
\(\chi_{507}(59,\cdot)\)
\(\chi_{507}(71,\cdot)\)
\(\chi_{507}(98,\cdot)\)
\(\chi_{507}(110,\cdot)\)
\(\chi_{507}(119,\cdot)\)
\(\chi_{507}(128,\cdot)\)
\(\chi_{507}(137,\cdot)\)
\(\chi_{507}(149,\cdot)\)
\(\chi_{507}(158,\cdot)\)
\(\chi_{507}(167,\cdot)\)
\(\chi_{507}(176,\cdot)\)
\(\chi_{507}(197,\cdot)\)
\(\chi_{507}(206,\cdot)\)
\(\chi_{507}(215,\cdot)\)
\(\chi_{507}(227,\cdot)\)
\(\chi_{507}(236,\cdot)\)
\(\chi_{507}(245,\cdot)\)
\(\chi_{507}(254,\cdot)\)
\(\chi_{507}(266,\cdot)\)
\(\chi_{507}(275,\cdot)\)
\(\chi_{507}(284,\cdot)\)
\(\chi_{507}(293,\cdot)\)
\(\chi_{507}(305,\cdot)\)
\(\chi_{507}(314,\cdot)\)
...
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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 ]
\((170,340)\) → \((-1,e\left(\frac{155}{156}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 507 }(254, a) \) |
\(1\) | \(1\) | \(e\left(\frac{77}{156}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{23}{52}\right)\) | \(e\left(\frac{49}{156}\right)\) | \(e\left(\frac{25}{52}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{131}{156}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{22}{39}\right)\) |
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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)
