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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2299, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([51,11]))
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pari:[g,chi] = znchar(Mod(1528,2299))
| Modulus: | \(2299\) | |
| Conductor: | \(2299\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(66\) |
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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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\(\chi_{2299}(65,\cdot)\)
\(\chi_{2299}(164,\cdot)\)
\(\chi_{2299}(274,\cdot)\)
\(\chi_{2299}(373,\cdot)\)
\(\chi_{2299}(582,\cdot)\)
\(\chi_{2299}(692,\cdot)\)
\(\chi_{2299}(791,\cdot)\)
\(\chi_{2299}(901,\cdot)\)
\(\chi_{2299}(1000,\cdot)\)
\(\chi_{2299}(1110,\cdot)\)
\(\chi_{2299}(1319,\cdot)\)
\(\chi_{2299}(1418,\cdot)\)
\(\chi_{2299}(1528,\cdot)\)
\(\chi_{2299}(1627,\cdot)\)
\(\chi_{2299}(1737,\cdot)\)
\(\chi_{2299}(1836,\cdot)\)
\(\chi_{2299}(1946,\cdot)\)
\(\chi_{2299}(2045,\cdot)\)
\(\chi_{2299}(2155,\cdot)\)
\(\chi_{2299}(2254,\cdot)\)
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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 ]
\((970,1332)\) → \((e\left(\frac{17}{22}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 2299 }(1528, a) \) |
\(1\) | \(1\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{1}{22}\right)\) |
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sage:chi.jacobi_sum(n)
