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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1045, base_ring=CyclotomicField(90))
M = H._module
chi = DirichletCharacter(H, M([45,18,50]))
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pari:[g,chi] = znchar(Mod(169,1045))
| Modulus: | \(1045\) | |
| Conductor: | \(1045\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(90\) |
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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
|
| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{1045}(4,\cdot)\)
\(\chi_{1045}(9,\cdot)\)
\(\chi_{1045}(104,\cdot)\)
\(\chi_{1045}(119,\cdot)\)
\(\chi_{1045}(169,\cdot)\)
\(\chi_{1045}(214,\cdot)\)
\(\chi_{1045}(234,\cdot)\)
\(\chi_{1045}(289,\cdot)\)
\(\chi_{1045}(339,\cdot)\)
\(\chi_{1045}(389,\cdot)\)
\(\chi_{1045}(434,\cdot)\)
\(\chi_{1045}(454,\cdot)\)
\(\chi_{1045}(499,\cdot)\)
\(\chi_{1045}(614,\cdot)\)
\(\chi_{1045}(669,\cdot)\)
\(\chi_{1045}(674,\cdot)\)
\(\chi_{1045}(709,\cdot)\)
\(\chi_{1045}(719,\cdot)\)
\(\chi_{1045}(764,\cdot)\)
\(\chi_{1045}(784,\cdot)\)
\(\chi_{1045}(834,\cdot)\)
\(\chi_{1045}(929,\cdot)\)
\(\chi_{1045}(994,\cdot)\)
\(\chi_{1045}(1004,\cdot)\)
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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 ]
\((837,761,496)\) → \((-1,e\left(\frac{1}{5}\right),e\left(\frac{5}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 1045 }(169, a) \) |
\(1\) | \(1\) | \(e\left(\frac{23}{90}\right)\) | \(e\left(\frac{29}{90}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{43}{90}\right)\) | \(e\left(\frac{22}{45}\right)\) |
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sage:chi.jacobi_sum(n)
