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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5328, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,9,30,7]))
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pari:[g,chi] = znchar(Mod(4235,5328))
| Modulus: | \(5328\) | |
| Conductor: | \(5328\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(36\) |
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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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{5328}(59,\cdot)\)
\(\chi_{5328}(131,\cdot)\)
\(\chi_{5328}(923,\cdot)\)
\(\chi_{5328}(1091,\cdot)\)
\(\chi_{5328}(2603,\cdot)\)
\(\chi_{5328}(2891,\cdot)\)
\(\chi_{5328}(3251,\cdot)\)
\(\chi_{5328}(3443,\cdot)\)
\(\chi_{5328}(3539,\cdot)\)
\(\chi_{5328}(4235,\cdot)\)
\(\chi_{5328}(4307,\cdot)\)
\(\chi_{5328}(5051,\cdot)\)
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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 ]
\((1999,1333,2369,1297)\) → \((-1,i,e\left(\frac{5}{6}\right),e\left(\frac{7}{36}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 5328 }(4235, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{12}\right)\) |
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sage:chi.jacobi_sum(n)
