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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1573, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([30,11]))
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pari:[g,chi] = znchar(Mod(1486,1573))
| Modulus: | \(1573\) | |
| Conductor: | \(1573\) |
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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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\(\chi_{1573}(23,\cdot)\)
\(\chi_{1573}(56,\cdot)\)
\(\chi_{1573}(166,\cdot)\)
\(\chi_{1573}(199,\cdot)\)
\(\chi_{1573}(309,\cdot)\)
\(\chi_{1573}(342,\cdot)\)
\(\chi_{1573}(452,\cdot)\)
\(\chi_{1573}(595,\cdot)\)
\(\chi_{1573}(628,\cdot)\)
\(\chi_{1573}(738,\cdot)\)
\(\chi_{1573}(771,\cdot)\)
\(\chi_{1573}(881,\cdot)\)
\(\chi_{1573}(914,\cdot)\)
\(\chi_{1573}(1024,\cdot)\)
\(\chi_{1573}(1057,\cdot)\)
\(\chi_{1573}(1167,\cdot)\)
\(\chi_{1573}(1200,\cdot)\)
\(\chi_{1573}(1310,\cdot)\)
\(\chi_{1573}(1343,\cdot)\)
\(\chi_{1573}(1486,\cdot)\)
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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 ]
\((365,1211)\) → \((e\left(\frac{5}{11}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 1573 }(1486, a) \) |
\(1\) | \(1\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) |
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sage:chi.jacobi_sum(n)
