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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1045, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([9,18,16]))
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pari:[g,chi] = znchar(Mod(142,1045))
| Modulus: | \(1045\) | |
| Conductor: | \(1045\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(36\) |
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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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\(\chi_{1045}(43,\cdot)\)
\(\chi_{1045}(142,\cdot)\)
\(\chi_{1045}(252,\cdot)\)
\(\chi_{1045}(263,\cdot)\)
\(\chi_{1045}(472,\cdot)\)
\(\chi_{1045}(538,\cdot)\)
\(\chi_{1045}(593,\cdot)\)
\(\chi_{1045}(747,\cdot)\)
\(\chi_{1045}(758,\cdot)\)
\(\chi_{1045}(802,\cdot)\)
\(\chi_{1045}(967,\cdot)\)
\(\chi_{1045}(978,\cdot)\)
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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 ]
\((837,761,496)\) → \((i,-1,e\left(\frac{4}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 1045 }(142, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) |
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sage:chi.jacobi_sum(n)
