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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([0,25,224]))
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gp:[g,chi] = znchar(Mod(309, 6400))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6400.309");
| 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;
|
| 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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\(\chi_{6400}(29,\cdot)\)
\(\chi_{6400}(69,\cdot)\)
\(\chi_{6400}(109,\cdot)\)
\(\chi_{6400}(189,\cdot)\)
\(\chi_{6400}(229,\cdot)\)
\(\chi_{6400}(269,\cdot)\)
\(\chi_{6400}(309,\cdot)\)
\(\chi_{6400}(389,\cdot)\)
\(\chi_{6400}(429,\cdot)\)
\(\chi_{6400}(469,\cdot)\)
\(\chi_{6400}(509,\cdot)\)
\(\chi_{6400}(589,\cdot)\)
\(\chi_{6400}(629,\cdot)\)
\(\chi_{6400}(669,\cdot)\)
\(\chi_{6400}(709,\cdot)\)
\(\chi_{6400}(789,\cdot)\)
\(\chi_{6400}(829,\cdot)\)
\(\chi_{6400}(869,\cdot)\)
\(\chi_{6400}(909,\cdot)\)
\(\chi_{6400}(989,\cdot)\)
\(\chi_{6400}(1029,\cdot)\)
\(\chi_{6400}(1069,\cdot)\)
\(\chi_{6400}(1109,\cdot)\)
\(\chi_{6400}(1189,\cdot)\)
\(\chi_{6400}(1229,\cdot)\)
\(\chi_{6400}(1269,\cdot)\)
\(\chi_{6400}(1309,\cdot)\)
\(\chi_{6400}(1389,\cdot)\)
\(\chi_{6400}(1429,\cdot)\)
\(\chi_{6400}(1469,\cdot)\)
...
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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()
|
\((4351,4101,5377)\) → \((1,e\left(\frac{5}{64}\right),e\left(\frac{7}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 6400 }(309, a) \) |
\(1\) | \(1\) | \(e\left(\frac{203}{320}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{43}{160}\right)\) | \(e\left(\frac{269}{320}\right)\) | \(e\left(\frac{311}{320}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{127}{320}\right)\) | \(e\left(\frac{293}{320}\right)\) | \(e\left(\frac{127}{160}\right)\) | \(e\left(\frac{289}{320}\right)\) |
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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)
