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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(20264, base_ring=CyclotomicField(296))
M = H._module
chi = DirichletCharacter(H, M([0,0,185,172]))
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gp:[g,chi] = znchar(Mod(2609, 20264))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("20264.2609");
| Modulus: | \(20264\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2533\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(296\) |
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sage:chi.multiplicative_order()
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gp:charorder(g,chi)
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magma:Order(chi);
|
| Real: | no |
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sage:chi.multiplicative_order() <= 2
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gp:charorder(g,chi) <= 2
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magma:Order(chi) le 2;
|
| Primitive: | no, induced from \(\chi_{2533}(76,\cdot)\) |
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sage:chi.is_primitive()
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gp:#znconreyconductor(g,chi)==1
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magma:IsPrimitive(chi);
|
| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{20264}(9,\cdot)\)
\(\chi_{20264}(121,\cdot)\)
\(\chi_{20264}(281,\cdot)\)
\(\chi_{20264}(417,\cdot)\)
\(\chi_{20264}(529,\cdot)\)
\(\chi_{20264}(665,\cdot)\)
\(\chi_{20264}(729,\cdot)\)
\(\chi_{20264}(865,\cdot)\)
\(\chi_{20264}(1097,\cdot)\)
\(\chi_{20264}(1345,\cdot)\)
\(\chi_{20264}(1409,\cdot)\)
\(\chi_{20264}(1681,\cdot)\)
\(\chi_{20264}(2049,\cdot)\)
\(\chi_{20264}(2321,\cdot)\)
\(\chi_{20264}(2497,\cdot)\)
\(\chi_{20264}(2609,\cdot)\)
\(\chi_{20264}(2633,\cdot)\)
\(\chi_{20264}(2729,\cdot)\)
\(\chi_{20264}(3041,\cdot)\)
\(\chi_{20264}(3113,\cdot)\)
\(\chi_{20264}(3153,\cdot)\)
\(\chi_{20264}(3249,\cdot)\)
\(\chi_{20264}(3273,\cdot)\)
\(\chi_{20264}(3313,\cdot)\)
\(\chi_{20264}(3449,\cdot)\)
\(\chi_{20264}(3545,\cdot)\)
\(\chi_{20264}(3585,\cdot)\)
\(\chi_{20264}(3697,\cdot)\)
\(\chi_{20264}(3793,\cdot)\)
\(\chi_{20264}(3857,\cdot)\)
...
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sage:chi.galois_orbit()
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gp:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
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magma:order := Order(chi);
{ chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
\((15199,10133,1193,17137)\) → \((1,1,e\left(\frac{5}{8}\right),e\left(\frac{43}{74}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 20264 }(2609, a) \) |
\(1\) | \(1\) | \(e\left(\frac{53}{296}\right)\) | \(e\left(\frac{165}{296}\right)\) | \(e\left(\frac{115}{296}\right)\) | \(e\left(\frac{53}{148}\right)\) | \(e\left(\frac{211}{296}\right)\) | \(e\left(\frac{11}{37}\right)\) | \(e\left(\frac{109}{148}\right)\) | \(e\left(\frac{83}{148}\right)\) | \(e\left(\frac{21}{37}\right)\) | \(e\left(\frac{171}{296}\right)\) |
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sage:chi(x)
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gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
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magma:chi(x)
