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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(416000, base_ring=CyclotomicField(320))
M = H._module
chi = DirichletCharacter(H, M([160,45,64,160]))
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gp:[g,chi] = znchar(Mod(23451, 416000))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("416000.23451");
| 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: | \(83200\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(320\) |
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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 \(\chi_{83200}(40091,\cdot)\) |
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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: | odd |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{416000}(51,\cdot)\)
\(\chi_{416000}(2651,\cdot)\)
\(\chi_{416000}(7851,\cdot)\)
\(\chi_{416000}(10451,\cdot)\)
\(\chi_{416000}(13051,\cdot)\)
\(\chi_{416000}(15651,\cdot)\)
\(\chi_{416000}(20851,\cdot)\)
\(\chi_{416000}(23451,\cdot)\)
\(\chi_{416000}(26051,\cdot)\)
\(\chi_{416000}(28651,\cdot)\)
\(\chi_{416000}(33851,\cdot)\)
\(\chi_{416000}(36451,\cdot)\)
\(\chi_{416000}(39051,\cdot)\)
\(\chi_{416000}(41651,\cdot)\)
\(\chi_{416000}(46851,\cdot)\)
\(\chi_{416000}(49451,\cdot)\)
\(\chi_{416000}(52051,\cdot)\)
\(\chi_{416000}(54651,\cdot)\)
\(\chi_{416000}(59851,\cdot)\)
\(\chi_{416000}(62451,\cdot)\)
\(\chi_{416000}(65051,\cdot)\)
\(\chi_{416000}(67651,\cdot)\)
\(\chi_{416000}(72851,\cdot)\)
\(\chi_{416000}(75451,\cdot)\)
\(\chi_{416000}(78051,\cdot)\)
\(\chi_{416000}(80651,\cdot)\)
\(\chi_{416000}(85851,\cdot)\)
\(\chi_{416000}(88451,\cdot)\)
\(\chi_{416000}(91051,\cdot)\)
\(\chi_{416000}(93651,\cdot)\)
...
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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: |
$\Q(\zeta_{320})$ |
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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 320 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((74751,266501,389377,64001)\) → \((-1,e\left(\frac{9}{64}\right),e\left(\frac{1}{5}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 416000 }(23451, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{263}{320}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{103}{160}\right)\) | \(e\left(\frac{49}{320}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{267}{320}\right)\) | \(e\left(\frac{73}{320}\right)\) | \(e\left(\frac{107}{160}\right)\) | \(e\left(\frac{149}{320}\right)\) | \(e\left(\frac{223}{320}\right)\) |
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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)
