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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([93,25]))
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gp:[g,chi] = znchar(Mod(654, 1625))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1625.654");
| 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;
|
| 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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\(\chi_{1625}(4,\cdot)\)
\(\chi_{1625}(69,\cdot)\)
\(\chi_{1625}(114,\cdot)\)
\(\chi_{1625}(134,\cdot)\)
\(\chi_{1625}(179,\cdot)\)
\(\chi_{1625}(244,\cdot)\)
\(\chi_{1625}(264,\cdot)\)
\(\chi_{1625}(309,\cdot)\)
\(\chi_{1625}(329,\cdot)\)
\(\chi_{1625}(394,\cdot)\)
\(\chi_{1625}(439,\cdot)\)
\(\chi_{1625}(459,\cdot)\)
\(\chi_{1625}(504,\cdot)\)
\(\chi_{1625}(569,\cdot)\)
\(\chi_{1625}(589,\cdot)\)
\(\chi_{1625}(634,\cdot)\)
\(\chi_{1625}(654,\cdot)\)
\(\chi_{1625}(719,\cdot)\)
\(\chi_{1625}(764,\cdot)\)
\(\chi_{1625}(784,\cdot)\)
\(\chi_{1625}(829,\cdot)\)
\(\chi_{1625}(894,\cdot)\)
\(\chi_{1625}(914,\cdot)\)
\(\chi_{1625}(959,\cdot)\)
\(\chi_{1625}(979,\cdot)\)
\(\chi_{1625}(1044,\cdot)\)
\(\chi_{1625}(1089,\cdot)\)
\(\chi_{1625}(1109,\cdot)\)
\(\chi_{1625}(1154,\cdot)\)
\(\chi_{1625}(1219,\cdot)\)
...
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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));
|
| Fixed field: |
Number field defined by a degree 150 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((1002,626)\) → \((e\left(\frac{31}{50}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 1625 }(654, a) \) |
\(1\) | \(1\) | \(e\left(\frac{59}{75}\right)\) | \(e\left(\frac{1}{150}\right)\) | \(e\left(\frac{43}{75}\right)\) | \(e\left(\frac{119}{150}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{1}{75}\right)\) | \(e\left(\frac{43}{150}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{8}{25}\right)\) |
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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)
