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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5625, base_ring=CyclotomicField(500))
M = H._module
chi = DirichletCharacter(H, M([250,317]))
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gp:[g,chi] = znchar(Mod(197, 5625))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5625.197");
| Modulus: | \(5625\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1875\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(500\) |
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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 \(\chi_{1875}(197,\cdot)\) |
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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: | yes |
| 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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\(\chi_{5625}(8,\cdot)\)
\(\chi_{5625}(17,\cdot)\)
\(\chi_{5625}(53,\cdot)\)
\(\chi_{5625}(62,\cdot)\)
\(\chi_{5625}(98,\cdot)\)
\(\chi_{5625}(152,\cdot)\)
\(\chi_{5625}(188,\cdot)\)
\(\chi_{5625}(197,\cdot)\)
\(\chi_{5625}(233,\cdot)\)
\(\chi_{5625}(242,\cdot)\)
\(\chi_{5625}(278,\cdot)\)
\(\chi_{5625}(287,\cdot)\)
\(\chi_{5625}(323,\cdot)\)
\(\chi_{5625}(377,\cdot)\)
\(\chi_{5625}(413,\cdot)\)
\(\chi_{5625}(422,\cdot)\)
\(\chi_{5625}(458,\cdot)\)
\(\chi_{5625}(467,\cdot)\)
\(\chi_{5625}(503,\cdot)\)
\(\chi_{5625}(512,\cdot)\)
\(\chi_{5625}(548,\cdot)\)
\(\chi_{5625}(602,\cdot)\)
\(\chi_{5625}(638,\cdot)\)
\(\chi_{5625}(647,\cdot)\)
\(\chi_{5625}(683,\cdot)\)
\(\chi_{5625}(692,\cdot)\)
\(\chi_{5625}(728,\cdot)\)
\(\chi_{5625}(737,\cdot)\)
\(\chi_{5625}(773,\cdot)\)
\(\chi_{5625}(827,\cdot)\)
...
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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: |
$\Q(\zeta_{500})$ |
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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 500 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((4376,1252)\) → \((-1,e\left(\frac{317}{500}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 5625 }(197, a) \) |
\(1\) | \(1\) | \(e\left(\frac{67}{500}\right)\) | \(e\left(\frac{67}{250}\right)\) | \(e\left(\frac{49}{100}\right)\) | \(e\left(\frac{201}{500}\right)\) | \(e\left(\frac{71}{250}\right)\) | \(e\left(\frac{63}{500}\right)\) | \(e\left(\frac{78}{125}\right)\) | \(e\left(\frac{67}{125}\right)\) | \(e\left(\frac{91}{500}\right)\) | \(e\left(\frac{3}{250}\right)\) |
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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)
