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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(836352, base_ring=CyclotomicField(10560))
M = H._module
chi = DirichletCharacter(H, M([0,4125,3520,4032]))
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gp:[g,chi] = znchar(Mod(37, 836352))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("836352.37");
| Modulus: | \(836352\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(278784\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(10560\) |
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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 \(\chi_{278784}(92965,\cdot)\) |
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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: | no |
| 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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\(\chi_{836352}(37,\cdot)\)
\(\chi_{836352}(181,\cdot)\)
\(\chi_{836352}(685,\cdot)\)
\(\chi_{836352}(829,\cdot)\)
\(\chi_{836352}(1477,\cdot)\)
\(\chi_{836352}(1549,\cdot)\)
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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: |
$\Q(\zeta_{10560})$ |
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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 10560 polynomial (not computed) |
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sage:chi.fixed_field()
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\((137215,561925,123905,781057)\) → \((1,e\left(\frac{25}{64}\right),e\left(\frac{1}{3}\right),e\left(\frac{21}{55}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 836352 }(37, a) \) |
\(1\) | \(1\) | \(e\left(\frac{3293}{10560}\right)\) | \(e\left(\frac{4817}{5280}\right)\) | \(e\left(\frac{6227}{10560}\right)\) | \(e\left(\frac{569}{880}\right)\) | \(e\left(\frac{2377}{3520}\right)\) | \(e\left(\frac{911}{1056}\right)\) | \(e\left(\frac{3293}{5280}\right)\) | \(e\left(\frac{9199}{10560}\right)\) | \(e\left(\frac{829}{1320}\right)\) | \(e\left(\frac{789}{3520}\right)\) |
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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)
