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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(228672, base_ring=CyclotomicField(1584))
M = H._module
chi = DirichletCharacter(H, M([792,495,0,760]))
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gp:[g,chi] = znchar(Mod(6283, 228672))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("228672.6283");
| Modulus: | \(228672\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(25408\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1584\) |
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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: | no, induced from \(\chi_{25408}(6283,\cdot)\) |
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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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\(\chi_{228672}(91,\cdot)\)
\(\chi_{228672}(307,\cdot)\)
\(\chi_{228672}(667,\cdot)\)
\(\chi_{228672}(1099,\cdot)\)
\(\chi_{228672}(1459,\cdot)\)
\(\chi_{228672}(1675,\cdot)\)
\(\chi_{228672}(2611,\cdot)\)
\(\chi_{228672}(2827,\cdot)\)
\(\chi_{228672}(3547,\cdot)\)
\(\chi_{228672}(3691,\cdot)\)
\(\chi_{228672}(4267,\cdot)\)
\(\chi_{228672}(4411,\cdot)\)
\(\chi_{228672}(4483,\cdot)\)
\(\chi_{228672}(4555,\cdot)\)
\(\chi_{228672}(5203,\cdot)\)
\(\chi_{228672}(5491,\cdot)\)
\(\chi_{228672}(5779,\cdot)\)
\(\chi_{228672}(5995,\cdot)\)
\(\chi_{228672}(6283,\cdot)\)
\(\chi_{228672}(6355,\cdot)\)
\(\chi_{228672}(6571,\cdot)\)
\(\chi_{228672}(7579,\cdot)\)
\(\chi_{228672}(7867,\cdot)\)
\(\chi_{228672}(8659,\cdot)\)
\(\chi_{228672}(9235,\cdot)\)
\(\chi_{228672}(10027,\cdot)\)
\(\chi_{228672}(11035,\cdot)\)
\(\chi_{228672}(11107,\cdot)\)
\(\chi_{228672}(11899,\cdot)\)
\(\chi_{228672}(12115,\cdot)\)
...
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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: |
$\Q(\zeta_{1584})$ |
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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 1584 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((21439,185797,25409,169921)\) → \((-1,e\left(\frac{5}{16}\right),1,e\left(\frac{95}{198}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 228672 }(6283, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1255}{1584}\right)\) | \(e\left(\frac{379}{792}\right)\) | \(e\left(\frac{371}{1584}\right)\) | \(e\left(\frac{233}{1584}\right)\) | \(e\left(\frac{109}{132}\right)\) | \(e\left(\frac{161}{1584}\right)\) | \(e\left(\frac{565}{792}\right)\) | \(e\left(\frac{463}{792}\right)\) | \(e\left(\frac{1013}{1584}\right)\) | \(e\left(\frac{5}{11}\right)\) |
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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)
