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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(836352, base_ring=CyclotomicField(1056))
M = H._module
chi = DirichletCharacter(H, M([0,891,176,768]))
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gp:[g,chi] = znchar(Mod(33353, 836352))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("836352.33353");
| Modulus: | \(836352\) |
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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: | no, induced from \(\chi_{139392}(78365,\cdot)\) |
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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: | 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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\(\chi_{836352}(89,\cdot)\)
\(\chi_{836352}(3257,\cdot)\)
\(\chi_{836352}(8009,\cdot)\)
\(\chi_{836352}(9593,\cdot)\)
\(\chi_{836352}(12761,\cdot)\)
\(\chi_{836352}(14345,\cdot)\)
\(\chi_{836352}(17513,\cdot)\)
\(\chi_{836352}(19097,\cdot)\)
\(\chi_{836352}(23849,\cdot)\)
\(\chi_{836352}(27017,\cdot)\)
\(\chi_{836352}(28601,\cdot)\)
\(\chi_{836352}(31769,\cdot)\)
\(\chi_{836352}(33353,\cdot)\)
\(\chi_{836352}(36521,\cdot)\)
\(\chi_{836352}(38105,\cdot)\)
\(\chi_{836352}(41273,\cdot)\)
\(\chi_{836352}(42857,\cdot)\)
\(\chi_{836352}(46025,\cdot)\)
\(\chi_{836352}(47609,\cdot)\)
\(\chi_{836352}(50777,\cdot)\)
\(\chi_{836352}(52361,\cdot)\)
\(\chi_{836352}(55529,\cdot)\)
\(\chi_{836352}(60281,\cdot)\)
\(\chi_{836352}(61865,\cdot)\)
\(\chi_{836352}(65033,\cdot)\)
\(\chi_{836352}(66617,\cdot)\)
\(\chi_{836352}(69785,\cdot)\)
\(\chi_{836352}(71369,\cdot)\)
\(\chi_{836352}(76121,\cdot)\)
\(\chi_{836352}(79289,\cdot)\)
...
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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()
|
\((137215,561925,123905,781057)\) → \((1,e\left(\frac{27}{32}\right),e\left(\frac{1}{6}\right),e\left(\frac{8}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 836352 }(33353, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{523}{1056}\right)\) | \(e\left(\frac{103}{528}\right)\) | \(e\left(\frac{469}{1056}\right)\) | \(e\left(\frac{67}{88}\right)\) | \(e\left(\frac{271}{352}\right)\) | \(e\left(\frac{293}{528}\right)\) | \(e\left(\frac{523}{528}\right)\) | \(e\left(\frac{329}{1056}\right)\) | \(e\left(\frac{83}{132}\right)\) | \(e\left(\frac{243}{352}\right)\) |
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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)
