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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8712, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,33,11,48]))
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pari:[g,chi] = znchar(Mod(6491,8712))
| Modulus: | \(8712\) | |
| Conductor: | \(8712\) |
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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
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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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\(\chi_{8712}(155,\cdot)\)
\(\chi_{8712}(419,\cdot)\)
\(\chi_{8712}(947,\cdot)\)
\(\chi_{8712}(1739,\cdot)\)
\(\chi_{8712}(2003,\cdot)\)
\(\chi_{8712}(2531,\cdot)\)
\(\chi_{8712}(2795,\cdot)\)
\(\chi_{8712}(3323,\cdot)\)
\(\chi_{8712}(3587,\cdot)\)
\(\chi_{8712}(4379,\cdot)\)
\(\chi_{8712}(4907,\cdot)\)
\(\chi_{8712}(5171,\cdot)\)
\(\chi_{8712}(5699,\cdot)\)
\(\chi_{8712}(5963,\cdot)\)
\(\chi_{8712}(6491,\cdot)\)
\(\chi_{8712}(6755,\cdot)\)
\(\chi_{8712}(7283,\cdot)\)
\(\chi_{8712}(7547,\cdot)\)
\(\chi_{8712}(8075,\cdot)\)
\(\chi_{8712}(8339,\cdot)\)
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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 ]
\((6535,4357,1937,5689)\) → \((-1,-1,e\left(\frac{1}{6}\right),e\left(\frac{8}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 8712 }(6491, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{9}{22}\right)\) |
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sage:chi.jacobi_sum(n)
