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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(41600, base_ring=CyclotomicField(160))
M = H._module
chi = DirichletCharacter(H, M([0,155,8,80]))
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gp:[g,chi] = znchar(Mod(77, 41600))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("41600.77");
| Modulus: | \(41600\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(41600\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(160\) |
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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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\(\chi_{41600}(77,\cdot)\)
\(\chi_{41600}(1013,\cdot)\)
\(\chi_{41600}(1117,\cdot)\)
\(\chi_{41600}(2053,\cdot)\)
\(\chi_{41600}(3197,\cdot)\)
\(\chi_{41600}(4133,\cdot)\)
\(\chi_{41600}(4237,\cdot)\)
\(\chi_{41600}(5173,\cdot)\)
\(\chi_{41600}(5277,\cdot)\)
\(\chi_{41600}(6213,\cdot)\)
\(\chi_{41600}(6317,\cdot)\)
\(\chi_{41600}(7253,\cdot)\)
\(\chi_{41600}(8397,\cdot)\)
\(\chi_{41600}(9333,\cdot)\)
\(\chi_{41600}(9437,\cdot)\)
\(\chi_{41600}(10373,\cdot)\)
\(\chi_{41600}(10477,\cdot)\)
\(\chi_{41600}(11413,\cdot)\)
\(\chi_{41600}(11517,\cdot)\)
\(\chi_{41600}(12453,\cdot)\)
\(\chi_{41600}(13597,\cdot)\)
\(\chi_{41600}(14533,\cdot)\)
\(\chi_{41600}(14637,\cdot)\)
\(\chi_{41600}(15573,\cdot)\)
\(\chi_{41600}(15677,\cdot)\)
\(\chi_{41600}(16613,\cdot)\)
\(\chi_{41600}(16717,\cdot)\)
\(\chi_{41600}(17653,\cdot)\)
\(\chi_{41600}(18797,\cdot)\)
\(\chi_{41600}(19733,\cdot)\)
...
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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: |
$\Q(\zeta_{160})$ |
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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 160 polynomial (not computed) |
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sage:chi.fixed_field()
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\((33151,16901,14977,22401)\) → \((1,e\left(\frac{31}{32}\right),e\left(\frac{1}{20}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 41600 }(77, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{41}{160}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{103}{160}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{109}{160}\right)\) | \(e\left(\frac{111}{160}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{123}{160}\right)\) | \(e\left(\frac{41}{160}\right)\) |
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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)
