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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(221, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([20,27]))
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pari:[g,chi] = znchar(Mod(201,221))
| Modulus: | \(221\) | |
| Conductor: | \(221\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(48\) |
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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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\(\chi_{221}(7,\cdot)\)
\(\chi_{221}(11,\cdot)\)
\(\chi_{221}(24,\cdot)\)
\(\chi_{221}(54,\cdot)\)
\(\chi_{221}(71,\cdot)\)
\(\chi_{221}(80,\cdot)\)
\(\chi_{221}(97,\cdot)\)
\(\chi_{221}(158,\cdot)\)
\(\chi_{221}(163,\cdot)\)
\(\chi_{221}(175,\cdot)\)
\(\chi_{221}(176,\cdot)\)
\(\chi_{221}(180,\cdot)\)
\(\chi_{221}(184,\cdot)\)
\(\chi_{221}(193,\cdot)\)
\(\chi_{221}(201,\cdot)\)
\(\chi_{221}(215,\cdot)\)
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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 ]
\((171,105)\) → \((e\left(\frac{5}{12}\right),e\left(\frac{9}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 221 }(201, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) |
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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)
