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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(58492, base_ring=CyclotomicField(696))
M = H._module
chi = DirichletCharacter(H, M([0,116,77]))
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gp:[g,chi] = znchar(Mod(2817, 58492))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("58492.2817");
| Modulus: | \(58492\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(14623\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(696\) |
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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 \(\chi_{14623}(2817,\cdot)\) |
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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: | 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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\(\chi_{58492}(33,\cdot)\)
\(\chi_{58492}(229,\cdot)\)
\(\chi_{58492}(257,\cdot)\)
\(\chi_{58492}(633,\cdot)\)
\(\chi_{58492}(773,\cdot)\)
\(\chi_{58492}(873,\cdot)\)
\(\chi_{58492}(885,\cdot)\)
\(\chi_{58492}(1361,\cdot)\)
\(\chi_{58492}(1725,\cdot)\)
\(\chi_{58492}(1825,\cdot)\)
\(\chi_{58492}(1937,\cdot)\)
\(\chi_{58492}(2273,\cdot)\)
\(\chi_{58492}(2453,\cdot)\)
\(\chi_{58492}(2693,\cdot)\)
\(\chi_{58492}(2817,\cdot)\)
\(\chi_{58492}(3293,\cdot)\)
\(\chi_{58492}(3405,\cdot)\)
\(\chi_{58492}(3545,\cdot)\)
\(\chi_{58492}(3561,\cdot)\)
\(\chi_{58492}(4245,\cdot)\)
\(\chi_{58492}(4357,\cdot)\)
\(\chi_{58492}(4497,\cdot)\)
\(\chi_{58492}(4721,\cdot)\)
\(\chi_{58492}(5001,\cdot)\)
\(\chi_{58492}(5213,\cdot)\)
\(\chi_{58492}(6221,\cdot)\)
\(\chi_{58492}(6305,\cdot)\)
\(\chi_{58492}(6333,\cdot)\)
\(\chi_{58492}(6401,\cdot)\)
\(\chi_{58492}(6653,\cdot)\)
...
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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: |
$\Q(\zeta_{696})$ |
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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 696 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((29247,50137,54321)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{77}{696}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 58492 }(2817, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{116}\right)\) | \(e\left(\frac{127}{348}\right)\) | \(e\left(\frac{19}{58}\right)\) | \(e\left(\frac{247}{696}\right)\) | \(e\left(\frac{233}{348}\right)\) | \(e\left(\frac{46}{87}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{595}{696}\right)\) | \(e\left(\frac{397}{696}\right)\) | \(e\left(\frac{127}{174}\right)\) |
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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)
