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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5082, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([0,220,216]))
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gp:[g,chi] = znchar(Mod(697, 5082))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5082.697");
| Modulus: | \(5082\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(847\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(165\) |
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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 \(\chi_{847}(697,\cdot)\) |
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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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\(\chi_{5082}(25,\cdot)\)
\(\chi_{5082}(37,\cdot)\)
\(\chi_{5082}(163,\cdot)\)
\(\chi_{5082}(235,\cdot)\)
\(\chi_{5082}(247,\cdot)\)
\(\chi_{5082}(289,\cdot)\)
\(\chi_{5082}(361,\cdot)\)
\(\chi_{5082}(445,\cdot)\)
\(\chi_{5082}(499,\cdot)\)
\(\chi_{5082}(625,\cdot)\)
\(\chi_{5082}(697,\cdot)\)
\(\chi_{5082}(709,\cdot)\)
\(\chi_{5082}(751,\cdot)\)
\(\chi_{5082}(823,\cdot)\)
\(\chi_{5082}(907,\cdot)\)
\(\chi_{5082}(949,\cdot)\)
\(\chi_{5082}(961,\cdot)\)
\(\chi_{5082}(1087,\cdot)\)
\(\chi_{5082}(1159,\cdot)\)
\(\chi_{5082}(1171,\cdot)\)
\(\chi_{5082}(1285,\cdot)\)
\(\chi_{5082}(1369,\cdot)\)
\(\chi_{5082}(1411,\cdot)\)
\(\chi_{5082}(1423,\cdot)\)
\(\chi_{5082}(1549,\cdot)\)
\(\chi_{5082}(1621,\cdot)\)
\(\chi_{5082}(1633,\cdot)\)
\(\chi_{5082}(1675,\cdot)\)
\(\chi_{5082}(1747,\cdot)\)
\(\chi_{5082}(1831,\cdot)\)
...
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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: |
$\Q(\zeta_{165})$ |
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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 165 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((3389,4357,2059)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{36}{55}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 5082 }(697, a) \) |
\(1\) | \(1\) | \(e\left(\frac{127}{165}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{122}{165}\right)\) | \(e\left(\frac{109}{165}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{89}{165}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{158}{165}\right)\) | \(e\left(\frac{136}{165}\right)\) | \(e\left(\frac{3}{55}\right)\) |
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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)
