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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(36864, base_ring=CyclotomicField(3072))
M = H._module
chi = DirichletCharacter(H, M([0,171,2048]))
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gp:[g,chi] = znchar(Mod(421, 36864))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("36864.421");
| Modulus: | \(36864\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(36864\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(3072\) |
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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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\(\chi_{36864}(13,\cdot)\)
\(\chi_{36864}(61,\cdot)\)
\(\chi_{36864}(85,\cdot)\)
\(\chi_{36864}(133,\cdot)\)
\(\chi_{36864}(157,\cdot)\)
\(\chi_{36864}(205,\cdot)\)
\(\chi_{36864}(229,\cdot)\)
\(\chi_{36864}(277,\cdot)\)
\(\chi_{36864}(301,\cdot)\)
\(\chi_{36864}(349,\cdot)\)
\(\chi_{36864}(373,\cdot)\)
\(\chi_{36864}(421,\cdot)\)
\(\chi_{36864}(445,\cdot)\)
\(\chi_{36864}(493,\cdot)\)
\(\chi_{36864}(517,\cdot)\)
\(\chi_{36864}(565,\cdot)\)
\(\chi_{36864}(589,\cdot)\)
\(\chi_{36864}(637,\cdot)\)
\(\chi_{36864}(661,\cdot)\)
\(\chi_{36864}(709,\cdot)\)
\(\chi_{36864}(733,\cdot)\)
\(\chi_{36864}(781,\cdot)\)
\(\chi_{36864}(805,\cdot)\)
\(\chi_{36864}(853,\cdot)\)
\(\chi_{36864}(877,\cdot)\)
\(\chi_{36864}(925,\cdot)\)
\(\chi_{36864}(949,\cdot)\)
\(\chi_{36864}(997,\cdot)\)
\(\chi_{36864}(1021,\cdot)\)
\(\chi_{36864}(1069,\cdot)\)
...
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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 };
\((8191,20485,4097)\) → \((1,e\left(\frac{57}{1024}\right),e\left(\frac{2}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 36864 }(421, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1195}{3072}\right)\) | \(e\left(\frac{631}{1536}\right)\) | \(e\left(\frac{1607}{3072}\right)\) | \(e\left(\frac{1189}{3072}\right)\) | \(e\left(\frac{239}{256}\right)\) | \(e\left(\frac{159}{1024}\right)\) | \(e\left(\frac{365}{1536}\right)\) | \(e\left(\frac{1195}{1536}\right)\) | \(e\left(\frac{809}{3072}\right)\) | \(e\left(\frac{59}{384}\right)\) |
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sage:chi(x)
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gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
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magma:chi(x)
