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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([40,33]))
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pari:[g,chi] = znchar(Mod(75,221))
| 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)
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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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\(\chi_{221}(10,\cdot)\)
\(\chi_{221}(23,\cdot)\)
\(\chi_{221}(56,\cdot)\)
\(\chi_{221}(62,\cdot)\)
\(\chi_{221}(75,\cdot)\)
\(\chi_{221}(82,\cdot)\)
\(\chi_{221}(88,\cdot)\)
\(\chi_{221}(95,\cdot)\)
\(\chi_{221}(108,\cdot)\)
\(\chi_{221}(114,\cdot)\)
\(\chi_{221}(147,\cdot)\)
\(\chi_{221}(160,\cdot)\)
\(\chi_{221}(173,\cdot)\)
\(\chi_{221}(192,\cdot)\)
\(\chi_{221}(199,\cdot)\)
\(\chi_{221}(218,\cdot)\)
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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 ]
\((171,105)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{11}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 221 }(75, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) |
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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)
