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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8112, base_ring=CyclotomicField(52))
M = H._module
chi = DirichletCharacter(H, M([26,13,26,36]))
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pari:[g,chi] = znchar(Mod(3563,8112))
| Modulus: | \(8112\) | |
| Conductor: | \(8112\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(52\) |
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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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\(\chi_{8112}(131,\cdot)\)
\(\chi_{8112}(443,\cdot)\)
\(\chi_{8112}(755,\cdot)\)
\(\chi_{8112}(1067,\cdot)\)
\(\chi_{8112}(1379,\cdot)\)
\(\chi_{8112}(2003,\cdot)\)
\(\chi_{8112}(2315,\cdot)\)
\(\chi_{8112}(2627,\cdot)\)
\(\chi_{8112}(2939,\cdot)\)
\(\chi_{8112}(3251,\cdot)\)
\(\chi_{8112}(3563,\cdot)\)
\(\chi_{8112}(3875,\cdot)\)
\(\chi_{8112}(4187,\cdot)\)
\(\chi_{8112}(4499,\cdot)\)
\(\chi_{8112}(4811,\cdot)\)
\(\chi_{8112}(5123,\cdot)\)
\(\chi_{8112}(5435,\cdot)\)
\(\chi_{8112}(6059,\cdot)\)
\(\chi_{8112}(6371,\cdot)\)
\(\chi_{8112}(6683,\cdot)\)
\(\chi_{8112}(6995,\cdot)\)
\(\chi_{8112}(7307,\cdot)\)
\(\chi_{8112}(7619,\cdot)\)
\(\chi_{8112}(7931,\cdot)\)
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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 ]
\((5071,6085,2705,3889)\) → \((-1,i,-1,e\left(\frac{9}{13}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 8112 }(3563, a) \) |
\(1\) | \(1\) | \(e\left(\frac{51}{52}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{29}{52}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(i\) | \(-1\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{49}{52}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{3}{52}\right)\) |
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sage:chi.jacobi_sum(n)
