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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(416000, base_ring=CyclotomicField(1600))
M = H._module
chi = DirichletCharacter(H, M([0,125,1072,0]))
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gp:[g,chi] = znchar(Mod(53, 416000))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("416000.53");
| Modulus: | \(416000\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(32000\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1600\) |
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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 \(\chi_{32000}(53,\cdot)\) |
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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: | odd |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{416000}(53,\cdot)\)
\(\chi_{416000}(1197,\cdot)\)
\(\chi_{416000}(2133,\cdot)\)
\(\chi_{416000}(2237,\cdot)\)
\(\chi_{416000}(3173,\cdot)\)
\(\chi_{416000}(3277,\cdot)\)
\(\chi_{416000}(4213,\cdot)\)
\(\chi_{416000}(4317,\cdot)\)
\(\chi_{416000}(5253,\cdot)\)
\(\chi_{416000}(6397,\cdot)\)
\(\chi_{416000}(7333,\cdot)\)
\(\chi_{416000}(7437,\cdot)\)
\(\chi_{416000}(8373,\cdot)\)
\(\chi_{416000}(8477,\cdot)\)
\(\chi_{416000}(9413,\cdot)\)
\(\chi_{416000}(9517,\cdot)\)
\(\chi_{416000}(10453,\cdot)\)
\(\chi_{416000}(11597,\cdot)\)
\(\chi_{416000}(12533,\cdot)\)
\(\chi_{416000}(12637,\cdot)\)
\(\chi_{416000}(13573,\cdot)\)
\(\chi_{416000}(13677,\cdot)\)
\(\chi_{416000}(14613,\cdot)\)
\(\chi_{416000}(14717,\cdot)\)
\(\chi_{416000}(15653,\cdot)\)
\(\chi_{416000}(16797,\cdot)\)
\(\chi_{416000}(17733,\cdot)\)
\(\chi_{416000}(17837,\cdot)\)
\(\chi_{416000}(18773,\cdot)\)
\(\chi_{416000}(18877,\cdot)\)
...
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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 };
| Field of values: |
$\Q(\zeta_{1600})$ |
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sage:CyclotomicField(chi.multiplicative_order())
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gp:nfinit(polcyclo(charorder(g,chi)))
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magma:CyclotomicField(Order(chi));
|
| Fixed field: |
Number field defined by a degree 1600 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((74751,266501,389377,64001)\) → \((1,e\left(\frac{5}{64}\right),e\left(\frac{67}{100}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 416000 }(53, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{679}{1600}\right)\) | \(e\left(\frac{117}{160}\right)\) | \(e\left(\frac{679}{800}\right)\) | \(e\left(\frac{897}{1600}\right)\) | \(e\left(\frac{39}{400}\right)\) | \(e\left(\frac{1371}{1600}\right)\) | \(e\left(\frac{249}{1600}\right)\) | \(e\left(\frac{691}{800}\right)\) | \(e\left(\frac{437}{1600}\right)\) | \(e\left(\frac{239}{1600}\right)\) |
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sage:chi(x) # x integer
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gp:chareval(g,chi,x) \\ x integer, value in Q/Z
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magma:chi(x)
