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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(41600, base_ring=CyclotomicField(480))
M = H._module
chi = DirichletCharacter(H, M([240,405,24,200]))
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gp:[g,chi] = znchar(Mod(227, 41600))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("41600.227");
| Modulus: | \(41600\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(41600\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(480\) |
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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);
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\(\chi_{41600}(67,\cdot)\)
\(\chi_{41600}(227,\cdot)\)
\(\chi_{41600}(683,\cdot)\)
\(\chi_{41600}(1267,\cdot)\)
\(\chi_{41600}(1723,\cdot)\)
\(\chi_{41600}(1883,\cdot)\)
\(\chi_{41600}(2147,\cdot)\)
\(\chi_{41600}(2763,\cdot)\)
\(\chi_{41600}(2923,\cdot)\)
\(\chi_{41600}(3187,\cdot)\)
\(\chi_{41600}(3347,\cdot)\)
\(\chi_{41600}(3803,\cdot)\)
\(\chi_{41600}(3963,\cdot)\)
\(\chi_{41600}(4227,\cdot)\)
\(\chi_{41600}(4387,\cdot)\)
\(\chi_{41600}(5003,\cdot)\)
\(\chi_{41600}(5267,\cdot)\)
\(\chi_{41600}(5427,\cdot)\)
\(\chi_{41600}(5883,\cdot)\)
\(\chi_{41600}(6467,\cdot)\)
\(\chi_{41600}(6923,\cdot)\)
\(\chi_{41600}(7083,\cdot)\)
\(\chi_{41600}(7347,\cdot)\)
\(\chi_{41600}(7963,\cdot)\)
\(\chi_{41600}(8123,\cdot)\)
\(\chi_{41600}(8387,\cdot)\)
\(\chi_{41600}(8547,\cdot)\)
\(\chi_{41600}(9003,\cdot)\)
\(\chi_{41600}(9163,\cdot)\)
\(\chi_{41600}(9427,\cdot)\)
...
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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: |
$\Q(\zeta_{480})$ |
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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 480 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((33151,16901,14977,22401)\) → \((-1,e\left(\frac{27}{32}\right),e\left(\frac{1}{20}\right),e\left(\frac{5}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 41600 }(227, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{23}{480}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{23}{240}\right)\) | \(e\left(\frac{449}{480}\right)\) | \(e\left(\frac{13}{120}\right)\) | \(e\left(\frac{427}{480}\right)\) | \(e\left(\frac{131}{160}\right)\) | \(e\left(\frac{7}{240}\right)\) | \(e\left(\frac{23}{160}\right)\) | \(e\left(\frac{263}{480}\right)\) |
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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)
