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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1037, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([15,38]))
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pari:[g,chi] = znchar(Mod(659,1037))
| Modulus: | \(1037\) | |
| Conductor: | \(1037\) |
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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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\(\chi_{1037}(4,\cdot)\)
\(\chi_{1037}(106,\cdot)\)
\(\chi_{1037}(293,\cdot)\)
\(\chi_{1037}(310,\cdot)\)
\(\chi_{1037}(344,\cdot)\)
\(\chi_{1037}(370,\cdot)\)
\(\chi_{1037}(412,\cdot)\)
\(\chi_{1037}(446,\cdot)\)
\(\chi_{1037}(463,\cdot)\)
\(\chi_{1037}(472,\cdot)\)
\(\chi_{1037}(659,\cdot)\)
\(\chi_{1037}(676,\cdot)\)
\(\chi_{1037}(710,\cdot)\)
\(\chi_{1037}(778,\cdot)\)
\(\chi_{1037}(812,\cdot)\)
\(\chi_{1037}(829,\cdot)\)
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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 ]
\((428,307)\) → \((i,e\left(\frac{19}{30}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 1037 }(659, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(i\) |
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sage:chi.jacobi_sum(n)
