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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4900, base_ring=CyclotomicField(420))
M = H._module
chi = DirichletCharacter(H, M([0,147,370]))
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gp:[g,chi] = znchar(Mod(1053, 4900))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4900.1053");
| Modulus: | \(4900\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1225\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(420\) |
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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 \(\chi_{1225}(1053,\cdot)\) |
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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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\(\chi_{4900}(17,\cdot)\)
\(\chi_{4900}(33,\cdot)\)
\(\chi_{4900}(73,\cdot)\)
\(\chi_{4900}(173,\cdot)\)
\(\chi_{4900}(213,\cdot)\)
\(\chi_{4900}(297,\cdot)\)
\(\chi_{4900}(353,\cdot)\)
\(\chi_{4900}(397,\cdot)\)
\(\chi_{4900}(437,\cdot)\)
\(\chi_{4900}(453,\cdot)\)
\(\chi_{4900}(537,\cdot)\)
\(\chi_{4900}(577,\cdot)\)
\(\chi_{4900}(633,\cdot)\)
\(\chi_{4900}(677,\cdot)\)
\(\chi_{4900}(733,\cdot)\)
\(\chi_{4900}(773,\cdot)\)
\(\chi_{4900}(817,\cdot)\)
\(\chi_{4900}(873,\cdot)\)
\(\chi_{4900}(997,\cdot)\)
\(\chi_{4900}(1013,\cdot)\)
\(\chi_{4900}(1053,\cdot)\)
\(\chi_{4900}(1137,\cdot)\)
\(\chi_{4900}(1153,\cdot)\)
\(\chi_{4900}(1237,\cdot)\)
\(\chi_{4900}(1277,\cdot)\)
\(\chi_{4900}(1333,\cdot)\)
\(\chi_{4900}(1377,\cdot)\)
\(\chi_{4900}(1417,\cdot)\)
\(\chi_{4900}(1433,\cdot)\)
\(\chi_{4900}(1473,\cdot)\)
...
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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: |
$\Q(\zeta_{420})$ |
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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 420 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((2451,1177,101)\) → \((1,e\left(\frac{7}{20}\right),e\left(\frac{37}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 4900 }(1053, a) \) |
\(1\) | \(1\) | \(e\left(\frac{139}{420}\right)\) | \(e\left(\frac{139}{210}\right)\) | \(e\left(\frac{88}{105}\right)\) | \(e\left(\frac{101}{140}\right)\) | \(e\left(\frac{241}{420}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{137}{420}\right)\) | \(e\left(\frac{139}{140}\right)\) | \(e\left(\frac{39}{70}\right)\) | \(e\left(\frac{29}{30}\right)\) |
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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)
