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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(355008, base_ring=CyclotomicField(14448))
M = H._module
chi = DirichletCharacter(H, M([0,903,7224,1544]))
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gp:[g,chi] = znchar(Mod(5, 355008))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("355008.5");
| Modulus: | \(355008\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(355008\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(14448\) |
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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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\(\chi_{355008}(5,\cdot)\)
\(\chi_{355008}(29,\cdot)\)
\(\chi_{355008}(77,\cdot)\)
\(\chi_{355008}(149,\cdot)\)
\(\chi_{355008}(245,\cdot)\)
\(\chi_{355008}(413,\cdot)\)
\(\chi_{355008}(485,\cdot)\)
\(\chi_{355008}(605,\cdot)\)
\(\chi_{355008}(749,\cdot)\)
\(\chi_{355008}(845,\cdot)\)
\(\chi_{355008}(893,\cdot)\)
\(\chi_{355008}(965,\cdot)\)
\(\chi_{355008}(1037,\cdot)\)
\(\chi_{355008}(1061,\cdot)\)
\(\chi_{355008}(1109,\cdot)\)
\(\chi_{355008}(1181,\cdot)\)
\(\chi_{355008}(1277,\cdot)\)
\(\chi_{355008}(1445,\cdot)\)
\(\chi_{355008}(1517,\cdot)\)
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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: |
$\Q(\zeta_{14448})$ |
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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 14448 polynomial (not computed) |
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sage:chi.fixed_field()
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\((321727,66565,118337,85057)\) → \((1,e\left(\frac{1}{16}\right),-1,e\left(\frac{193}{1806}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 355008 }(5, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2711}{14448}\right)\) | \(e\left(\frac{833}{1032}\right)\) | \(e\left(\frac{3001}{4816}\right)\) | \(e\left(\frac{1129}{14448}\right)\) | \(e\left(\frac{2299}{3612}\right)\) | \(e\left(\frac{251}{336}\right)\) | \(e\left(\frac{4309}{7224}\right)\) | \(e\left(\frac{2711}{7224}\right)\) | \(e\left(\frac{8221}{14448}\right)\) | \(e\left(\frac{1165}{1806}\right)\) |
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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)
