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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(58492, base_ring=CyclotomicField(1044))
M = H._module
chi = DirichletCharacter(H, M([0,522,269]))
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gp:[g,chi] = znchar(Mod(405, 58492))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("58492.405");
| 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: | \(1044\) |
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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 \(\chi_{14623}(405,\cdot)\) |
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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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\(\chi_{58492}(13,\cdot)\)
\(\chi_{58492}(153,\cdot)\)
\(\chi_{58492}(209,\cdot)\)
\(\chi_{58492}(293,\cdot)\)
\(\chi_{58492}(377,\cdot)\)
\(\chi_{58492}(405,\cdot)\)
\(\chi_{58492}(713,\cdot)\)
\(\chi_{58492}(1105,\cdot)\)
\(\chi_{58492}(1917,\cdot)\)
\(\chi_{58492}(2113,\cdot)\)
\(\chi_{58492}(2141,\cdot)\)
\(\chi_{58492}(2169,\cdot)\)
\(\chi_{58492}(2225,\cdot)\)
\(\chi_{58492}(2281,\cdot)\)
\(\chi_{58492}(2505,\cdot)\)
\(\chi_{58492}(2701,\cdot)\)
\(\chi_{58492}(2729,\cdot)\)
\(\chi_{58492}(2757,\cdot)\)
\(\chi_{58492}(2785,\cdot)\)
\(\chi_{58492}(2925,\cdot)\)
\(\chi_{58492}(3177,\cdot)\)
\(\chi_{58492}(3261,\cdot)\)
\(\chi_{58492}(3625,\cdot)\)
\(\chi_{58492}(3709,\cdot)\)
\(\chi_{58492}(3793,\cdot)\)
\(\chi_{58492}(4017,\cdot)\)
\(\chi_{58492}(4129,\cdot)\)
\(\chi_{58492}(4409,\cdot)\)
\(\chi_{58492}(4437,\cdot)\)
\(\chi_{58492}(4885,\cdot)\)
...
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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: |
$\Q(\zeta_{1044})$ |
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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 1044 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((29247,50137,54321)\) → \((1,-1,e\left(\frac{269}{1044}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 58492 }(405, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{70}{261}\right)\) | \(e\left(\frac{13}{261}\right)\) | \(e\left(\frac{140}{261}\right)\) | \(e\left(\frac{887}{1044}\right)\) | \(e\left(\frac{46}{261}\right)\) | \(e\left(\frac{83}{261}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{83}{348}\right)\) | \(e\left(\frac{275}{348}\right)\) | \(e\left(\frac{26}{261}\right)\) |
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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)
