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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([0,315,288,440]))
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gp:[g,chi] = znchar(Mod(8821, 41600))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("41600.8821");
| 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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\(\chi_{41600}(461,\cdot)\)
\(\chi_{41600}(661,\cdot)\)
\(\chi_{41600}(1341,\cdot)\)
\(\chi_{41600}(1541,\cdot)\)
\(\chi_{41600}(2381,\cdot)\)
\(\chi_{41600}(2541,\cdot)\)
\(\chi_{41600}(2581,\cdot)\)
\(\chi_{41600}(2741,\cdot)\)
\(\chi_{41600}(3421,\cdot)\)
\(\chi_{41600}(3581,\cdot)\)
\(\chi_{41600}(3621,\cdot)\)
\(\chi_{41600}(3781,\cdot)\)
\(\chi_{41600}(4461,\cdot)\)
\(\chi_{41600}(4621,\cdot)\)
\(\chi_{41600}(4661,\cdot)\)
\(\chi_{41600}(4821,\cdot)\)
\(\chi_{41600}(5661,\cdot)\)
\(\chi_{41600}(5861,\cdot)\)
\(\chi_{41600}(6541,\cdot)\)
\(\chi_{41600}(6741,\cdot)\)
\(\chi_{41600}(7581,\cdot)\)
\(\chi_{41600}(7741,\cdot)\)
\(\chi_{41600}(7781,\cdot)\)
\(\chi_{41600}(7941,\cdot)\)
\(\chi_{41600}(8621,\cdot)\)
\(\chi_{41600}(8781,\cdot)\)
\(\chi_{41600}(8821,\cdot)\)
\(\chi_{41600}(8981,\cdot)\)
\(\chi_{41600}(9661,\cdot)\)
\(\chi_{41600}(9821,\cdot)\)
...
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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()
|
\((33151,16901,14977,22401)\) → \((1,e\left(\frac{21}{32}\right),e\left(\frac{3}{5}\right),e\left(\frac{11}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 41600 }(8821, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{401}{480}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{161}{240}\right)\) | \(e\left(\frac{383}{480}\right)\) | \(e\left(\frac{1}{120}\right)\) | \(e\left(\frac{229}{480}\right)\) | \(e\left(\frac{77}{160}\right)\) | \(e\left(\frac{229}{240}\right)\) | \(e\left(\frac{81}{160}\right)\) | \(e\left(\frac{281}{480}\right)\) |
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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)
