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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1640, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,20,10,13]))
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pari:[g,chi] = znchar(Mod(147,1640))
| Modulus: | \(1640\) | |
| Conductor: | \(1640\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(40\) |
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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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{1640}(67,\cdot)\)
\(\chi_{1640}(147,\cdot)\)
\(\chi_{1640}(227,\cdot)\)
\(\chi_{1640}(347,\cdot)\)
\(\chi_{1640}(403,\cdot)\)
\(\chi_{1640}(427,\cdot)\)
\(\chi_{1640}(507,\cdot)\)
\(\chi_{1640}(563,\cdot)\)
\(\chi_{1640}(603,\cdot)\)
\(\chi_{1640}(643,\cdot)\)
\(\chi_{1640}(867,\cdot)\)
\(\chi_{1640}(1243,\cdot)\)
\(\chi_{1640}(1283,\cdot)\)
\(\chi_{1640}(1323,\cdot)\)
\(\chi_{1640}(1347,\cdot)\)
\(\chi_{1640}(1483,\cdot)\)
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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 ]
\((1231,821,657,1441)\) → \((-1,-1,i,e\left(\frac{13}{40}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 1640 }(147, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(i\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) |
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sage:chi.jacobi_sum(n)
