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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(416000, base_ring=CyclotomicField(240))
M = H._module
chi = DirichletCharacter(H, M([0,45,168,200]))
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gp:[g,chi] = znchar(Mod(52049, 416000))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("416000.52049");
| Modulus: | \(416000\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(20800\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(240\) |
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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 \(\chi_{20800}(13309,\cdot)\) |
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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: | even |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{416000}(49,\cdot)\)
\(\chi_{416000}(849,\cdot)\)
\(\chi_{416000}(10449,\cdot)\)
\(\chi_{416000}(20849,\cdot)\)
\(\chi_{416000}(21649,\cdot)\)
\(\chi_{416000}(32049,\cdot)\)
\(\chi_{416000}(41649,\cdot)\)
\(\chi_{416000}(42449,\cdot)\)
\(\chi_{416000}(52049,\cdot)\)
\(\chi_{416000}(52849,\cdot)\)
\(\chi_{416000}(62449,\cdot)\)
\(\chi_{416000}(72849,\cdot)\)
\(\chi_{416000}(73649,\cdot)\)
\(\chi_{416000}(84049,\cdot)\)
\(\chi_{416000}(93649,\cdot)\)
\(\chi_{416000}(94449,\cdot)\)
\(\chi_{416000}(104049,\cdot)\)
\(\chi_{416000}(104849,\cdot)\)
\(\chi_{416000}(114449,\cdot)\)
\(\chi_{416000}(124849,\cdot)\)
\(\chi_{416000}(125649,\cdot)\)
\(\chi_{416000}(136049,\cdot)\)
\(\chi_{416000}(145649,\cdot)\)
\(\chi_{416000}(146449,\cdot)\)
\(\chi_{416000}(156049,\cdot)\)
\(\chi_{416000}(156849,\cdot)\)
\(\chi_{416000}(166449,\cdot)\)
\(\chi_{416000}(176849,\cdot)\)
\(\chi_{416000}(177649,\cdot)\)
\(\chi_{416000}(188049,\cdot)\)
...
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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: |
$\Q(\zeta_{240})$ |
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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 240 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((74751,266501,389377,64001)\) → \((1,e\left(\frac{3}{16}\right),e\left(\frac{7}{10}\right),e\left(\frac{5}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 416000 }(52049, a) \) |
\(1\) | \(1\) | \(e\left(\frac{191}{240}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{71}{120}\right)\) | \(e\left(\frac{233}{240}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{19}{240}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{79}{120}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{191}{240}\right)\) |
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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)
