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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([0,27,6,31]))
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pari:[g,chi] = znchar(Mod(1613,5328))
| 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
|
| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{5328}(461,\cdot)\)
\(\chi_{5328}(533,\cdot)\)
\(\chi_{5328}(869,\cdot)\)
\(\chi_{5328}(1613,\cdot)\)
\(\chi_{5328}(1685,\cdot)\)
\(\chi_{5328}(2381,\cdot)\)
\(\chi_{5328}(2477,\cdot)\)
\(\chi_{5328}(2669,\cdot)\)
\(\chi_{5328}(3029,\cdot)\)
\(\chi_{5328}(3317,\cdot)\)
\(\chi_{5328}(4829,\cdot)\)
\(\chi_{5328}(4997,\cdot)\)
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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 ]
\((1999,1333,2369,1297)\) → \((1,-i,e\left(\frac{1}{6}\right),e\left(\frac{31}{36}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 5328 }(1613, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(-i\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(i\) | \(e\left(\frac{7}{9}\right)\) | \(-1\) | \(e\left(\frac{1}{12}\right)\) |
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sage:chi.jacobi_sum(n)
