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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5225, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([63,144,170]))
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gp:[g,chi] = znchar(Mod(1378, 5225))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5225.1378");
| Modulus: | \(5225\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(5225\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(180\) |
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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: | 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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\(\chi_{5225}(3,\cdot)\)
\(\chi_{5225}(148,\cdot)\)
\(\chi_{5225}(262,\cdot)\)
\(\chi_{5225}(433,\cdot)\)
\(\chi_{5225}(553,\cdot)\)
\(\chi_{5225}(642,\cdot)\)
\(\chi_{5225}(698,\cdot)\)
\(\chi_{5225}(713,\cdot)\)
\(\chi_{5225}(812,\cdot)\)
\(\chi_{5225}(983,\cdot)\)
\(\chi_{5225}(1192,\cdot)\)
\(\chi_{5225}(1248,\cdot)\)
\(\chi_{5225}(1362,\cdot)\)
\(\chi_{5225}(1378,\cdot)\)
\(\chi_{5225}(1402,\cdot)\)
\(\chi_{5225}(1523,\cdot)\)
\(\chi_{5225}(1533,\cdot)\)
\(\chi_{5225}(1637,\cdot)\)
\(\chi_{5225}(1742,\cdot)\)
\(\chi_{5225}(1808,\cdot)\)
\(\chi_{5225}(1952,\cdot)\)
\(\chi_{5225}(1972,\cdot)\)
\(\chi_{5225}(2017,\cdot)\)
\(\chi_{5225}(2073,\cdot)\)
\(\chi_{5225}(2187,\cdot)\)
\(\chi_{5225}(2358,\cdot)\)
\(\chi_{5225}(2502,\cdot)\)
\(\chi_{5225}(2522,\cdot)\)
\(\chi_{5225}(2567,\cdot)\)
\(\chi_{5225}(2777,\cdot)\)
...
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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: |
$\Q(\zeta_{180})$ |
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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 180 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((2927,2851,4676)\) → \((e\left(\frac{7}{20}\right),e\left(\frac{4}{5}\right),e\left(\frac{17}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 5225 }(1378, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{180}\right)\) | \(e\left(\frac{23}{180}\right)\) | \(e\left(\frac{17}{90}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{23}{90}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{31}{180}\right)\) | \(e\left(\frac{1}{9}\right)\) |
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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)
