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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6400, base_ring=CyclotomicField(320))
M = H._module
chi = DirichletCharacter(H, M([0,205,64]))
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gp:[g,chi] = znchar(Mod(741, 6400))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6400.741");
| Modulus: | \(6400\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(6400\) |
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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;
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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);
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| 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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\(\chi_{6400}(21,\cdot)\)
\(\chi_{6400}(61,\cdot)\)
\(\chi_{6400}(141,\cdot)\)
\(\chi_{6400}(181,\cdot)\)
\(\chi_{6400}(221,\cdot)\)
\(\chi_{6400}(261,\cdot)\)
\(\chi_{6400}(341,\cdot)\)
\(\chi_{6400}(381,\cdot)\)
\(\chi_{6400}(421,\cdot)\)
\(\chi_{6400}(461,\cdot)\)
\(\chi_{6400}(541,\cdot)\)
\(\chi_{6400}(581,\cdot)\)
\(\chi_{6400}(621,\cdot)\)
\(\chi_{6400}(661,\cdot)\)
\(\chi_{6400}(741,\cdot)\)
\(\chi_{6400}(781,\cdot)\)
\(\chi_{6400}(821,\cdot)\)
\(\chi_{6400}(861,\cdot)\)
\(\chi_{6400}(941,\cdot)\)
\(\chi_{6400}(981,\cdot)\)
\(\chi_{6400}(1021,\cdot)\)
\(\chi_{6400}(1061,\cdot)\)
\(\chi_{6400}(1141,\cdot)\)
\(\chi_{6400}(1181,\cdot)\)
\(\chi_{6400}(1221,\cdot)\)
\(\chi_{6400}(1261,\cdot)\)
\(\chi_{6400}(1341,\cdot)\)
\(\chi_{6400}(1381,\cdot)\)
\(\chi_{6400}(1421,\cdot)\)
\(\chi_{6400}(1461,\cdot)\)
...
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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));
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| Fixed field: |
Number field defined by a degree 320 polynomial (not computed) |
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sage:chi.fixed_field()
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\((4351,4101,5377)\) → \((1,e\left(\frac{41}{64}\right),e\left(\frac{1}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 6400 }(741, a) \) |
\(1\) | \(1\) | \(e\left(\frac{263}{320}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{103}{160}\right)\) | \(e\left(\frac{209}{320}\right)\) | \(e\left(\frac{291}{320}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{107}{320}\right)\) | \(e\left(\frac{73}{320}\right)\) | \(e\left(\frac{27}{160}\right)\) | \(e\left(\frac{149}{320}\right)\) |
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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)
