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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(416000, base_ring=CyclotomicField(2400))
M = H._module
chi = DirichletCharacter(H, M([0,2325,1056,200]))
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gp:[g,chi] = znchar(Mod(41, 416000))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("416000.41");
| 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: | \(208000\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(2400\) |
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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 \(\chi_{208000}(110541,\cdot)\) |
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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: | odd |
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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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\(\chi_{416000}(41,\cdot)\)
\(\chi_{416000}(761,\cdot)\)
\(\chi_{416000}(921,\cdot)\)
\(\chi_{416000}(2121,\cdot)\)
\(\chi_{416000}(2281,\cdot)\)
\(\chi_{416000}(2841,\cdot)\)
\(\chi_{416000}(4361,\cdot)\)
\(\chi_{416000}(4921,\cdot)\)
\(\chi_{416000}(5081,\cdot)\)
\(\chi_{416000}(6281,\cdot)\)
\(\chi_{416000}(6441,\cdot)\)
\(\chi_{416000}(7161,\cdot)\)
\(\chi_{416000}(8361,\cdot)\)
\(\chi_{416000}(8521,\cdot)\)
\(\chi_{416000}(9081,\cdot)\)
\(\chi_{416000}(9241,\cdot)\)
\(\chi_{416000}(10441,\cdot)\)
\(\chi_{416000}(11161,\cdot)\)
\(\chi_{416000}(11321,\cdot)\)
\(\chi_{416000}(12521,\cdot)\)
\(\chi_{416000}(12681,\cdot)\)
\(\chi_{416000}(13241,\cdot)\)
\(\chi_{416000}(14761,\cdot)\)
\(\chi_{416000}(15321,\cdot)\)
\(\chi_{416000}(15481,\cdot)\)
\(\chi_{416000}(16681,\cdot)\)
\(\chi_{416000}(16841,\cdot)\)
\(\chi_{416000}(17561,\cdot)\)
\(\chi_{416000}(18761,\cdot)\)
\(\chi_{416000}(18921,\cdot)\)
...
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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: |
$\Q(\zeta_{2400})$ |
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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 2400 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((74751,266501,389377,64001)\) → \((1,e\left(\frac{31}{32}\right),e\left(\frac{11}{25}\right),e\left(\frac{1}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 416000 }(41, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{767}{2400}\right)\) | \(e\left(\frac{1}{240}\right)\) | \(e\left(\frac{767}{1200}\right)\) | \(e\left(\frac{881}{2400}\right)\) | \(e\left(\frac{247}{600}\right)\) | \(e\left(\frac{1483}{2400}\right)\) | \(e\left(\frac{259}{800}\right)\) | \(e\left(\frac{43}{1200}\right)\) | \(e\left(\frac{767}{800}\right)\) | \(e\left(\frac{1847}{2400}\right)\) |
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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)
