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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(58492, base_ring=CyclotomicField(348))
M = H._module
chi = DirichletCharacter(H, M([0,58,165]))
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gp:[g,chi] = znchar(Mod(45, 58492))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("58492.45");
| 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: | \(348\) |
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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 \(\chi_{14623}(45,\cdot)\) |
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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);
|
\(\chi_{58492}(45,\cdot)\)
\(\chi_{58492}(61,\cdot)\)
\(\chi_{58492}(325,\cdot)\)
\(\chi_{58492}(649,\cdot)\)
\(\chi_{58492}(1265,\cdot)\)
\(\chi_{58492}(1601,\cdot)\)
\(\chi_{58492}(1909,\cdot)\)
\(\chi_{58492}(2637,\cdot)\)
\(\chi_{58492}(3853,\cdot)\)
\(\chi_{58492}(4133,\cdot)\)
\(\chi_{58492}(5253,\cdot)\)
\(\chi_{58492}(5617,\cdot)\)
\(\chi_{58492}(6177,\cdot)\)
\(\chi_{58492}(6389,\cdot)\)
\(\chi_{58492}(6473,\cdot)\)
\(\chi_{58492}(6541,\cdot)\)
\(\chi_{58492}(7565,\cdot)\)
\(\chi_{58492}(7845,\cdot)\)
\(\chi_{58492}(8417,\cdot)\)
\(\chi_{58492}(8797,\cdot)\)
\(\chi_{58492}(9005,\cdot)\)
\(\chi_{58492}(9469,\cdot)\)
\(\chi_{58492}(9621,\cdot)\)
\(\chi_{58492}(9957,\cdot)\)
\(\chi_{58492}(10085,\cdot)\)
\(\chi_{58492}(10265,\cdot)\)
\(\chi_{58492}(10993,\cdot)\)
\(\chi_{58492}(11541,\cdot)\)
\(\chi_{58492}(14745,\cdot)\)
\(\chi_{58492}(14829,\cdot)\)
...
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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: |
$\Q(\zeta_{348})$ |
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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 348 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((29247,50137,54321)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{55}{116}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 58492 }(45, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{87}\right)\) | \(e\left(\frac{47}{87}\right)\) | \(e\left(\frac{2}{87}\right)\) | \(e\left(\frac{115}{348}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{173}{348}\right)\) | \(e\left(\frac{221}{348}\right)\) | \(e\left(\frac{7}{87}\right)\) |
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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)
