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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(83200, base_ring=CyclotomicField(320))
M = H._module
chi = DirichletCharacter(H, M([160,175,192,160]))
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gp:[g,chi] = znchar(Mod(13571, 83200))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("83200.13571");
| Modulus: | \(83200\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(83200\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(320\) |
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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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\(\chi_{83200}(571,\cdot)\)
\(\chi_{83200}(1091,\cdot)\)
\(\chi_{83200}(1611,\cdot)\)
\(\chi_{83200}(2131,\cdot)\)
\(\chi_{83200}(3171,\cdot)\)
\(\chi_{83200}(3691,\cdot)\)
\(\chi_{83200}(4211,\cdot)\)
\(\chi_{83200}(4731,\cdot)\)
\(\chi_{83200}(5771,\cdot)\)
\(\chi_{83200}(6291,\cdot)\)
\(\chi_{83200}(6811,\cdot)\)
\(\chi_{83200}(7331,\cdot)\)
\(\chi_{83200}(8371,\cdot)\)
\(\chi_{83200}(8891,\cdot)\)
\(\chi_{83200}(9411,\cdot)\)
\(\chi_{83200}(9931,\cdot)\)
\(\chi_{83200}(10971,\cdot)\)
\(\chi_{83200}(11491,\cdot)\)
\(\chi_{83200}(12011,\cdot)\)
\(\chi_{83200}(12531,\cdot)\)
\(\chi_{83200}(13571,\cdot)\)
\(\chi_{83200}(14091,\cdot)\)
\(\chi_{83200}(14611,\cdot)\)
\(\chi_{83200}(15131,\cdot)\)
\(\chi_{83200}(16171,\cdot)\)
\(\chi_{83200}(16691,\cdot)\)
\(\chi_{83200}(17211,\cdot)\)
\(\chi_{83200}(17731,\cdot)\)
\(\chi_{83200}(18771,\cdot)\)
\(\chi_{83200}(19291,\cdot)\)
...
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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: |
$\Q(\zeta_{320})$ |
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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 320 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((74751,16901,56577,64001)\) → \((-1,e\left(\frac{35}{64}\right),e\left(\frac{3}{5}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 83200 }(13571, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{269}{320}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{109}{160}\right)\) | \(e\left(\frac{27}{320}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{121}{320}\right)\) | \(e\left(\frac{99}{320}\right)\) | \(e\left(\frac{121}{160}\right)\) | \(e\left(\frac{167}{320}\right)\) | \(e\left(\frac{149}{320}\right)\) |
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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)
