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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8000, base_ring=CyclotomicField(400))
M = H._module
chi = DirichletCharacter(H, M([200,225,392]))
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gp:[g,chi] = znchar(Mod(219, 8000))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8000.219");
| Modulus: | \(8000\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(8000\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(400\) |
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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: | odd |
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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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\(\chi_{8000}(19,\cdot)\)
\(\chi_{8000}(59,\cdot)\)
\(\chi_{8000}(139,\cdot)\)
\(\chi_{8000}(179,\cdot)\)
\(\chi_{8000}(219,\cdot)\)
\(\chi_{8000}(259,\cdot)\)
\(\chi_{8000}(339,\cdot)\)
\(\chi_{8000}(379,\cdot)\)
\(\chi_{8000}(419,\cdot)\)
\(\chi_{8000}(459,\cdot)\)
\(\chi_{8000}(539,\cdot)\)
\(\chi_{8000}(579,\cdot)\)
\(\chi_{8000}(619,\cdot)\)
\(\chi_{8000}(659,\cdot)\)
\(\chi_{8000}(739,\cdot)\)
\(\chi_{8000}(779,\cdot)\)
\(\chi_{8000}(819,\cdot)\)
\(\chi_{8000}(859,\cdot)\)
\(\chi_{8000}(939,\cdot)\)
\(\chi_{8000}(979,\cdot)\)
\(\chi_{8000}(1019,\cdot)\)
\(\chi_{8000}(1059,\cdot)\)
\(\chi_{8000}(1139,\cdot)\)
\(\chi_{8000}(1179,\cdot)\)
\(\chi_{8000}(1219,\cdot)\)
\(\chi_{8000}(1259,\cdot)\)
\(\chi_{8000}(1339,\cdot)\)
\(\chi_{8000}(1379,\cdot)\)
\(\chi_{8000}(1419,\cdot)\)
\(\chi_{8000}(1459,\cdot)\)
...
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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: |
$\Q(\zeta_{400})$ |
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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 400 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((2751,2501,5377)\) → \((-1,e\left(\frac{9}{16}\right),e\left(\frac{49}{50}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 8000 }(219, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{19}{400}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{19}{200}\right)\) | \(e\left(\frac{317}{400}\right)\) | \(e\left(\frac{263}{400}\right)\) | \(e\left(\frac{29}{100}\right)\) | \(e\left(\frac{31}{400}\right)\) | \(e\left(\frac{189}{400}\right)\) | \(e\left(\frac{151}{200}\right)\) | \(e\left(\frac{57}{400}\right)\) |
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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)
