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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(147, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,10]))
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pari:[g,chi] = znchar(Mod(53,147))
| Modulus: | \(147\) | |
| Conductor: | \(147\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(42\) |
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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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{147}(2,\cdot)\)
\(\chi_{147}(11,\cdot)\)
\(\chi_{147}(23,\cdot)\)
\(\chi_{147}(32,\cdot)\)
\(\chi_{147}(44,\cdot)\)
\(\chi_{147}(53,\cdot)\)
\(\chi_{147}(65,\cdot)\)
\(\chi_{147}(74,\cdot)\)
\(\chi_{147}(86,\cdot)\)
\(\chi_{147}(95,\cdot)\)
\(\chi_{147}(107,\cdot)\)
\(\chi_{147}(137,\cdot)\)
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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 ]
\((50,52)\) → \((-1,e\left(\frac{5}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 147 }(53, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{1}{3}\right)\) |
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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)
