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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2675, base_ring=CyclotomicField(530))
M = H._module
chi = DirichletCharacter(H, M([159,470]))
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gp:[g,chi] = znchar(Mod(239, 2675))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2675.239");
| Modulus: | \(2675\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2675\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(530\) |
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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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\(\chi_{2675}(4,\cdot)\)
\(\chi_{2675}(9,\cdot)\)
\(\chi_{2675}(14,\cdot)\)
\(\chi_{2675}(19,\cdot)\)
\(\chi_{2675}(29,\cdot)\)
\(\chi_{2675}(34,\cdot)\)
\(\chi_{2675}(39,\cdot)\)
\(\chi_{2675}(44,\cdot)\)
\(\chi_{2675}(64,\cdot)\)
\(\chi_{2675}(69,\cdot)\)
\(\chi_{2675}(79,\cdot)\)
\(\chi_{2675}(89,\cdot)\)
\(\chi_{2675}(119,\cdot)\)
\(\chi_{2675}(134,\cdot)\)
\(\chi_{2675}(144,\cdot)\)
\(\chi_{2675}(154,\cdot)\)
\(\chi_{2675}(159,\cdot)\)
\(\chi_{2675}(164,\cdot)\)
\(\chi_{2675}(169,\cdot)\)
\(\chi_{2675}(194,\cdot)\)
\(\chi_{2675}(209,\cdot)\)
\(\chi_{2675}(239,\cdot)\)
\(\chi_{2675}(244,\cdot)\)
\(\chi_{2675}(254,\cdot)\)
\(\chi_{2675}(289,\cdot)\)
\(\chi_{2675}(304,\cdot)\)
\(\chi_{2675}(314,\cdot)\)
\(\chi_{2675}(319,\cdot)\)
\(\chi_{2675}(334,\cdot)\)
\(\chi_{2675}(344,\cdot)\)
...
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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: |
$\Q(\zeta_{265})$ |
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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 530 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((1927,751)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{47}{53}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 2675 }(239, a) \) |
\(1\) | \(1\) | \(e\left(\frac{99}{530}\right)\) | \(e\left(\frac{93}{530}\right)\) | \(e\left(\frac{99}{265}\right)\) | \(e\left(\frac{96}{265}\right)\) | \(e\left(\frac{67}{106}\right)\) | \(e\left(\frac{297}{530}\right)\) | \(e\left(\frac{93}{265}\right)\) | \(e\left(\frac{82}{265}\right)\) | \(e\left(\frac{291}{530}\right)\) | \(e\left(\frac{61}{530}\right)\) |
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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)
