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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(139392, base_ring=CyclotomicField(1056))
M = H._module
chi = DirichletCharacter(H, M([0,957,176,288]))
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gp:[g,chi] = znchar(Mod(12629, 139392))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("139392.12629");
| Modulus: | \(139392\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(139392\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1056\) |
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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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\(\chi_{139392}(221,\cdot)\)
\(\chi_{139392}(749,\cdot)\)
\(\chi_{139392}(1013,\cdot)\)
\(\chi_{139392}(1541,\cdot)\)
\(\chi_{139392}(1805,\cdot)\)
\(\chi_{139392}(2333,\cdot)\)
\(\chi_{139392}(2597,\cdot)\)
\(\chi_{139392}(3125,\cdot)\)
\(\chi_{139392}(3917,\cdot)\)
\(\chi_{139392}(4181,\cdot)\)
\(\chi_{139392}(4709,\cdot)\)
\(\chi_{139392}(4973,\cdot)\)
\(\chi_{139392}(5501,\cdot)\)
\(\chi_{139392}(5765,\cdot)\)
\(\chi_{139392}(6557,\cdot)\)
\(\chi_{139392}(7085,\cdot)\)
\(\chi_{139392}(7349,\cdot)\)
\(\chi_{139392}(7877,\cdot)\)
\(\chi_{139392}(8141,\cdot)\)
\(\chi_{139392}(8669,\cdot)\)
\(\chi_{139392}(8933,\cdot)\)
\(\chi_{139392}(9461,\cdot)\)
\(\chi_{139392}(9725,\cdot)\)
\(\chi_{139392}(10253,\cdot)\)
\(\chi_{139392}(10517,\cdot)\)
\(\chi_{139392}(11045,\cdot)\)
\(\chi_{139392}(11309,\cdot)\)
\(\chi_{139392}(11837,\cdot)\)
\(\chi_{139392}(12629,\cdot)\)
\(\chi_{139392}(12893,\cdot)\)
...
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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: |
$\Q(\zeta_{1056})$ |
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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 1056 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((137215,4357,123905,84097)\) → \((1,e\left(\frac{29}{32}\right),e\left(\frac{1}{6}\right),e\left(\frac{3}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 139392 }(12629, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{973}{1056}\right)\) | \(e\left(\frac{337}{528}\right)\) | \(e\left(\frac{499}{1056}\right)\) | \(e\left(\frac{21}{88}\right)\) | \(e\left(\frac{169}{352}\right)\) | \(e\left(\frac{323}{528}\right)\) | \(e\left(\frac{445}{528}\right)\) | \(e\left(\frac{287}{1056}\right)\) | \(e\left(\frac{5}{132}\right)\) | \(e\left(\frac{197}{352}\right)\) |
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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)
