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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(262144, base_ring=CyclotomicField(2048))
M = H._module
chi = DirichletCharacter(H, M([0,1241]))
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gp:[g,chi] = znchar(Mod(897, 262144))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("262144.897");
| Modulus: | \(262144\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(8192\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(2048\) |
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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 \(\chi_{8192}(1317,\cdot)\) |
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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);
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\(\chi_{262144}(129,\cdot)\)
\(\chi_{262144}(385,\cdot)\)
\(\chi_{262144}(641,\cdot)\)
\(\chi_{262144}(897,\cdot)\)
\(\chi_{262144}(1153,\cdot)\)
\(\chi_{262144}(1409,\cdot)\)
\(\chi_{262144}(1665,\cdot)\)
\(\chi_{262144}(1921,\cdot)\)
\(\chi_{262144}(2177,\cdot)\)
\(\chi_{262144}(2433,\cdot)\)
\(\chi_{262144}(2689,\cdot)\)
\(\chi_{262144}(2945,\cdot)\)
\(\chi_{262144}(3201,\cdot)\)
\(\chi_{262144}(3457,\cdot)\)
\(\chi_{262144}(3713,\cdot)\)
\(\chi_{262144}(3969,\cdot)\)
\(\chi_{262144}(4225,\cdot)\)
\(\chi_{262144}(4481,\cdot)\)
\(\chi_{262144}(4737,\cdot)\)
\(\chi_{262144}(4993,\cdot)\)
\(\chi_{262144}(5249,\cdot)\)
\(\chi_{262144}(5505,\cdot)\)
\(\chi_{262144}(5761,\cdot)\)
\(\chi_{262144}(6017,\cdot)\)
\(\chi_{262144}(6273,\cdot)\)
\(\chi_{262144}(6529,\cdot)\)
\(\chi_{262144}(6785,\cdot)\)
\(\chi_{262144}(7041,\cdot)\)
\(\chi_{262144}(7297,\cdot)\)
\(\chi_{262144}(7553,\cdot)\)
...
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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 };
\((262143,5)\) → \((1,e\left(\frac{1241}{2048}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 262144 }(897, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1067}{2048}\right)\) | \(e\left(\frac{1241}{2048}\right)\) | \(e\left(\frac{157}{1024}\right)\) | \(e\left(\frac{43}{1024}\right)\) | \(e\left(\frac{1165}{2048}\right)\) | \(e\left(\frac{1431}{2048}\right)\) | \(e\left(\frac{65}{512}\right)\) | \(e\left(\frac{335}{512}\right)\) | \(e\left(\frac{1791}{2048}\right)\) | \(e\left(\frac{1381}{2048}\right)\) |
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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)
