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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5733, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([14,22,14]))
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pari:[g,chi] = znchar(Mod(3721,5733))
| Modulus: | \(5733\) | |
| Conductor: | \(5733\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(21\) |
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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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\(\chi_{5733}(445,\cdot)\)
\(\chi_{5733}(646,\cdot)\)
\(\chi_{5733}(1264,\cdot)\)
\(\chi_{5733}(1465,\cdot)\)
\(\chi_{5733}(2083,\cdot)\)
\(\chi_{5733}(2902,\cdot)\)
\(\chi_{5733}(3103,\cdot)\)
\(\chi_{5733}(3721,\cdot)\)
\(\chi_{5733}(3922,\cdot)\)
\(\chi_{5733}(4540,\cdot)\)
\(\chi_{5733}(4741,\cdot)\)
\(\chi_{5733}(5560,\cdot)\)
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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 ]
\((2549,1522,5293)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{11}{21}\right),e\left(\frac{1}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(16\) | \(17\) | \(19\) | \(20\) |
| \( \chi_{ 5733 }(3721, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(1\) | \(e\left(\frac{3}{7}\right)\) |
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sage:chi.jacobi_sum(n)
