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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1625, base_ring=CyclotomicField(150))
M = H._module
chi = DirichletCharacter(H, M([51,50]))
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gp:[g,chi] = znchar(Mod(809, 1625))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1625.809");
| Modulus: | \(1625\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1625\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(150\) |
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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;
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| 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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\(\chi_{1625}(9,\cdot)\)
\(\chi_{1625}(29,\cdot)\)
\(\chi_{1625}(94,\cdot)\)
\(\chi_{1625}(139,\cdot)\)
\(\chi_{1625}(159,\cdot)\)
\(\chi_{1625}(204,\cdot)\)
\(\chi_{1625}(269,\cdot)\)
\(\chi_{1625}(289,\cdot)\)
\(\chi_{1625}(334,\cdot)\)
\(\chi_{1625}(354,\cdot)\)
\(\chi_{1625}(419,\cdot)\)
\(\chi_{1625}(464,\cdot)\)
\(\chi_{1625}(484,\cdot)\)
\(\chi_{1625}(529,\cdot)\)
\(\chi_{1625}(594,\cdot)\)
\(\chi_{1625}(614,\cdot)\)
\(\chi_{1625}(659,\cdot)\)
\(\chi_{1625}(679,\cdot)\)
\(\chi_{1625}(744,\cdot)\)
\(\chi_{1625}(789,\cdot)\)
\(\chi_{1625}(809,\cdot)\)
\(\chi_{1625}(854,\cdot)\)
\(\chi_{1625}(919,\cdot)\)
\(\chi_{1625}(939,\cdot)\)
\(\chi_{1625}(984,\cdot)\)
\(\chi_{1625}(1004,\cdot)\)
\(\chi_{1625}(1069,\cdot)\)
\(\chi_{1625}(1114,\cdot)\)
\(\chi_{1625}(1134,\cdot)\)
\(\chi_{1625}(1179,\cdot)\)
...
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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: |
$\Q(\zeta_{75})$ |
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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));
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| Fixed field: |
Number field defined by a degree 150 polynomial (not computed) |
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sage:chi.fixed_field()
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\((1002,626)\) → \((e\left(\frac{17}{50}\right),e\left(\frac{1}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 1625 }(809, a) \) |
\(1\) | \(1\) | \(e\left(\frac{101}{150}\right)\) | \(e\left(\frac{107}{150}\right)\) | \(e\left(\frac{26}{75}\right)\) | \(e\left(\frac{29}{75}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{32}{75}\right)\) | \(e\left(\frac{13}{75}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{6}{25}\right)\) |
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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)
