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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9025, base_ring=CyclotomicField(1710))
M = H._module
chi = DirichletCharacter(H, M([1368,265]))
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gp:[g,chi] = znchar(Mod(86, 9025))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9025.86");
| Modulus: | \(9025\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(9025\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1710\) |
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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: | 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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\(\chi_{9025}(21,\cdot)\)
\(\chi_{9025}(41,\cdot)\)
\(\chi_{9025}(71,\cdot)\)
\(\chi_{9025}(86,\cdot)\)
\(\chi_{9025}(91,\cdot)\)
\(\chi_{9025}(136,\cdot)\)
\(\chi_{9025}(146,\cdot)\)
\(\chi_{9025}(166,\cdot)\)
\(\chi_{9025}(181,\cdot)\)
\(\chi_{9025}(186,\cdot)\)
\(\chi_{9025}(211,\cdot)\)
\(\chi_{9025}(231,\cdot)\)
\(\chi_{9025}(241,\cdot)\)
\(\chi_{9025}(261,\cdot)\)
\(\chi_{9025}(281,\cdot)\)
\(\chi_{9025}(306,\cdot)\)
\(\chi_{9025}(336,\cdot)\)
\(\chi_{9025}(356,\cdot)\)
\(\chi_{9025}(371,\cdot)\)
\(\chi_{9025}(421,\cdot)\)
\(\chi_{9025}(431,\cdot)\)
\(\chi_{9025}(466,\cdot)\)
\(\chi_{9025}(471,\cdot)\)
\(\chi_{9025}(496,\cdot)\)
\(\chi_{9025}(516,\cdot)\)
\(\chi_{9025}(546,\cdot)\)
\(\chi_{9025}(561,\cdot)\)
\(\chi_{9025}(566,\cdot)\)
\(\chi_{9025}(591,\cdot)\)
\(\chi_{9025}(611,\cdot)\)
...
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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: |
$\Q(\zeta_{855})$ |
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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 1710 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((5777,3251)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{53}{342}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 9025 }(86, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1633}{1710}\right)\) | \(e\left(\frac{241}{1710}\right)\) | \(e\left(\frac{778}{855}\right)\) | \(e\left(\frac{82}{855}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{493}{570}\right)\) | \(e\left(\frac{241}{855}\right)\) | \(e\left(\frac{173}{285}\right)\) | \(e\left(\frac{29}{570}\right)\) | \(e\left(\frac{317}{1710}\right)\) |
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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)
