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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1045, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([45,54,40]))
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pari:[g,chi] = znchar(Mod(1018,1045))
| Modulus: | \(1045\) | |
| Conductor: | \(1045\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(60\) |
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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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\(\chi_{1045}(7,\cdot)\)
\(\chi_{1045}(68,\cdot)\)
\(\chi_{1045}(83,\cdot)\)
\(\chi_{1045}(178,\cdot)\)
\(\chi_{1045}(182,\cdot)\)
\(\chi_{1045}(277,\cdot)\)
\(\chi_{1045}(292,\cdot)\)
\(\chi_{1045}(387,\cdot)\)
\(\chi_{1045}(448,\cdot)\)
\(\chi_{1045}(558,\cdot)\)
\(\chi_{1045}(657,\cdot)\)
\(\chi_{1045}(733,\cdot)\)
\(\chi_{1045}(767,\cdot)\)
\(\chi_{1045}(843,\cdot)\)
\(\chi_{1045}(942,\cdot)\)
\(\chi_{1045}(1018,\cdot)\)
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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 ]
\((837,761,496)\) → \((-i,e\left(\frac{9}{10}\right),e\left(\frac{2}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 1045 }(1018, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(-i\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) |
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sage:chi.jacobi_sum(n)
