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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([0,975,1872,800]))
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gp:[g,chi] = znchar(Mod(10169, 416000))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("416000.10169");
| 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 \(\chi_{208000}(107669,\cdot)\) |
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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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\(\chi_{416000}(9,\cdot)\)
\(\chi_{416000}(809,\cdot)\)
\(\chi_{416000}(2089,\cdot)\)
\(\chi_{416000}(2889,\cdot)\)
\(\chi_{416000}(3129,\cdot)\)
\(\chi_{416000}(3929,\cdot)\)
\(\chi_{416000}(4169,\cdot)\)
\(\chi_{416000}(4969,\cdot)\)
\(\chi_{416000}(5209,\cdot)\)
\(\chi_{416000}(6009,\cdot)\)
\(\chi_{416000}(7289,\cdot)\)
\(\chi_{416000}(8089,\cdot)\)
\(\chi_{416000}(8329,\cdot)\)
\(\chi_{416000}(9129,\cdot)\)
\(\chi_{416000}(9369,\cdot)\)
\(\chi_{416000}(10169,\cdot)\)
\(\chi_{416000}(10409,\cdot)\)
\(\chi_{416000}(11209,\cdot)\)
\(\chi_{416000}(12489,\cdot)\)
\(\chi_{416000}(13289,\cdot)\)
\(\chi_{416000}(13529,\cdot)\)
\(\chi_{416000}(14329,\cdot)\)
\(\chi_{416000}(14569,\cdot)\)
\(\chi_{416000}(15369,\cdot)\)
\(\chi_{416000}(15609,\cdot)\)
\(\chi_{416000}(16409,\cdot)\)
\(\chi_{416000}(17689,\cdot)\)
\(\chi_{416000}(18489,\cdot)\)
\(\chi_{416000}(18729,\cdot)\)
\(\chi_{416000}(19529,\cdot)\)
...
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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()
|
\((74751,266501,389377,64001)\) → \((1,e\left(\frac{13}{32}\right),e\left(\frac{39}{50}\right),e\left(\frac{1}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 416000 }(10169, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{2400}\right)\) | \(e\left(\frac{7}{240}\right)\) | \(e\left(\frac{29}{1200}\right)\) | \(e\left(\frac{347}{2400}\right)\) | \(e\left(\frac{589}{600}\right)\) | \(e\left(\frac{121}{2400}\right)\) | \(e\left(\frac{33}{800}\right)\) | \(e\left(\frac{241}{1200}\right)\) | \(e\left(\frac{29}{800}\right)\) | \(e\left(\frac{1589}{2400}\right)\) |
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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)
