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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5184, base_ring=CyclotomicField(144))
M = H._module
chi = DirichletCharacter(H, M([72,117,56]))
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gp:[g,chi] = znchar(Mod(2411, 5184))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5184.2411");
| Modulus: | \(5184\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1728\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(144\) |
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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 \(\chi_{1728}(1451,\cdot)\) |
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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: | no |
| 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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\(\chi_{5184}(35,\cdot)\)
\(\chi_{5184}(179,\cdot)\)
\(\chi_{5184}(251,\cdot)\)
\(\chi_{5184}(395,\cdot)\)
\(\chi_{5184}(467,\cdot)\)
\(\chi_{5184}(611,\cdot)\)
\(\chi_{5184}(683,\cdot)\)
\(\chi_{5184}(827,\cdot)\)
\(\chi_{5184}(899,\cdot)\)
\(\chi_{5184}(1043,\cdot)\)
\(\chi_{5184}(1115,\cdot)\)
\(\chi_{5184}(1259,\cdot)\)
\(\chi_{5184}(1331,\cdot)\)
\(\chi_{5184}(1475,\cdot)\)
\(\chi_{5184}(1547,\cdot)\)
\(\chi_{5184}(1691,\cdot)\)
\(\chi_{5184}(1763,\cdot)\)
\(\chi_{5184}(1907,\cdot)\)
\(\chi_{5184}(1979,\cdot)\)
\(\chi_{5184}(2123,\cdot)\)
\(\chi_{5184}(2195,\cdot)\)
\(\chi_{5184}(2339,\cdot)\)
\(\chi_{5184}(2411,\cdot)\)
\(\chi_{5184}(2555,\cdot)\)
\(\chi_{5184}(2627,\cdot)\)
\(\chi_{5184}(2771,\cdot)\)
\(\chi_{5184}(2843,\cdot)\)
\(\chi_{5184}(2987,\cdot)\)
\(\chi_{5184}(3059,\cdot)\)
\(\chi_{5184}(3203,\cdot)\)
...
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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: |
$\Q(\zeta_{144})$ |
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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 144 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((2431,325,1217)\) → \((-1,e\left(\frac{13}{16}\right),e\left(\frac{7}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 5184 }(2411, a) \) |
\(1\) | \(1\) | \(e\left(\frac{109}{144}\right)\) | \(e\left(\frac{61}{72}\right)\) | \(e\left(\frac{89}{144}\right)\) | \(e\left(\frac{43}{144}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{11}{72}\right)\) | \(e\left(\frac{37}{72}\right)\) | \(e\left(\frac{47}{144}\right)\) | \(e\left(\frac{7}{9}\right)\) |
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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)
