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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(49, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([38]))
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pari:[g,chi] = znchar(Mod(23,49))
| Modulus: | \(49\) | |
| Conductor: | \(49\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(21\) |
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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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{49}(2,\cdot)\)
\(\chi_{49}(4,\cdot)\)
\(\chi_{49}(9,\cdot)\)
\(\chi_{49}(11,\cdot)\)
\(\chi_{49}(16,\cdot)\)
\(\chi_{49}(23,\cdot)\)
\(\chi_{49}(25,\cdot)\)
\(\chi_{49}(32,\cdot)\)
\(\chi_{49}(37,\cdot)\)
\(\chi_{49}(39,\cdot)\)
\(\chi_{49}(44,\cdot)\)
\(\chi_{49}(46,\cdot)\)
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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 ]
\(3\) → \(e\left(\frac{19}{21}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 49 }(23, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) |
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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)
