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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(21296, base_ring=CyclotomicField(242))
M = H._module
chi = DirichletCharacter(H, M([0,0,84]))
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gp:[g,chi] = znchar(Mod(5457, 21296))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("21296.5457");
| Modulus: | \(21296\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1331\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(121\) |
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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 \(\chi_{1331}(133,\cdot)\) |
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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: | no |
| 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);
|
\(\chi_{21296}(177,\cdot)\)
\(\chi_{21296}(353,\cdot)\)
\(\chi_{21296}(529,\cdot)\)
\(\chi_{21296}(705,\cdot)\)
\(\chi_{21296}(881,\cdot)\)
\(\chi_{21296}(1057,\cdot)\)
\(\chi_{21296}(1233,\cdot)\)
\(\chi_{21296}(1409,\cdot)\)
\(\chi_{21296}(1585,\cdot)\)
\(\chi_{21296}(1761,\cdot)\)
\(\chi_{21296}(2113,\cdot)\)
\(\chi_{21296}(2289,\cdot)\)
\(\chi_{21296}(2465,\cdot)\)
\(\chi_{21296}(2641,\cdot)\)
\(\chi_{21296}(2817,\cdot)\)
\(\chi_{21296}(2993,\cdot)\)
\(\chi_{21296}(3169,\cdot)\)
\(\chi_{21296}(3345,\cdot)\)
\(\chi_{21296}(3521,\cdot)\)
\(\chi_{21296}(3697,\cdot)\)
\(\chi_{21296}(4049,\cdot)\)
\(\chi_{21296}(4225,\cdot)\)
\(\chi_{21296}(4401,\cdot)\)
\(\chi_{21296}(4577,\cdot)\)
\(\chi_{21296}(4753,\cdot)\)
\(\chi_{21296}(4929,\cdot)\)
\(\chi_{21296}(5105,\cdot)\)
\(\chi_{21296}(5281,\cdot)\)
\(\chi_{21296}(5457,\cdot)\)
\(\chi_{21296}(5633,\cdot)\)
...
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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 };
\((13311,15973,6657)\) → \((1,1,e\left(\frac{42}{121}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 21296 }(5457, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{105}{121}\right)\) | \(e\left(\frac{41}{121}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{106}{121}\right)\) | \(e\left(\frac{39}{121}\right)\) | \(e\left(\frac{23}{121}\right)\) | \(e\left(\frac{54}{121}\right)\) | \(e\left(\frac{96}{121}\right)\) | \(e\left(\frac{80}{121}\right)\) |
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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)
