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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([800,75,752,1200]))
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gp:[g,chi] = znchar(Mod(3203, 416000))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("416000.3203");
| 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: | \(416000\) |
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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: | yes |
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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);
|
\(\chi_{416000}(83,\cdot)\)
\(\chi_{416000}(827,\cdot)\)
\(\chi_{416000}(1123,\cdot)\)
\(\chi_{416000}(1867,\cdot)\)
\(\chi_{416000}(2163,\cdot)\)
\(\chi_{416000}(3203,\cdot)\)
\(\chi_{416000}(3947,\cdot)\)
\(\chi_{416000}(4987,\cdot)\)
\(\chi_{416000}(5283,\cdot)\)
\(\chi_{416000}(6027,\cdot)\)
\(\chi_{416000}(6323,\cdot)\)
\(\chi_{416000}(7067,\cdot)\)
\(\chi_{416000}(7363,\cdot)\)
\(\chi_{416000}(8403,\cdot)\)
\(\chi_{416000}(9147,\cdot)\)
\(\chi_{416000}(10187,\cdot)\)
\(\chi_{416000}(10483,\cdot)\)
\(\chi_{416000}(11227,\cdot)\)
\(\chi_{416000}(11523,\cdot)\)
\(\chi_{416000}(12267,\cdot)\)
\(\chi_{416000}(12563,\cdot)\)
\(\chi_{416000}(13603,\cdot)\)
\(\chi_{416000}(14347,\cdot)\)
\(\chi_{416000}(15387,\cdot)\)
\(\chi_{416000}(15683,\cdot)\)
\(\chi_{416000}(16427,\cdot)\)
\(\chi_{416000}(16723,\cdot)\)
\(\chi_{416000}(17467,\cdot)\)
\(\chi_{416000}(17763,\cdot)\)
\(\chi_{416000}(18803,\cdot)\)
...
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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()
|
\((74751,266501,389377,64001)\) → \((-1,e\left(\frac{3}{64}\right),e\left(\frac{47}{100}\right),-i)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 416000 }(3203, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{689}{1600}\right)\) | \(e\left(\frac{27}{160}\right)\) | \(e\left(\frac{689}{800}\right)\) | \(e\left(\frac{727}{1600}\right)\) | \(e\left(\frac{49}{400}\right)\) | \(e\left(\frac{1261}{1600}\right)\) | \(e\left(\frac{959}{1600}\right)\) | \(e\left(\frac{181}{800}\right)\) | \(e\left(\frac{467}{1600}\right)\) | \(e\left(\frac{1449}{1600}\right)\) |
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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)
