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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(355008, base_ring=CyclotomicField(1032))
M = H._module
chi = DirichletCharacter(H, M([516,387,0,340]))
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gp:[g,chi] = znchar(Mod(2359, 355008))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("355008.2359");
| Modulus: | \(355008\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(59168\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1032\) |
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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 \(\chi_{59168}(24547,\cdot)\) |
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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: | no |
| 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);
|
\(\chi_{355008}(7,\cdot)\)
\(\chi_{355008}(295,\cdot)\)
\(\chi_{355008}(2071,\cdot)\)
\(\chi_{355008}(2359,\cdot)\)
\(\chi_{355008}(4135,\cdot)\)
\(\chi_{355008}(4423,\cdot)\)
\(\chi_{355008}(6199,\cdot)\)
\(\chi_{355008}(6487,\cdot)\)
\(\chi_{355008}(8263,\cdot)\)
\(\chi_{355008}(8551,\cdot)\)
\(\chi_{355008}(10327,\cdot)\)
\(\chi_{355008}(10615,\cdot)\)
\(\chi_{355008}(12391,\cdot)\)
\(\chi_{355008}(12679,\cdot)\)
\(\chi_{355008}(14455,\cdot)\)
\(\chi_{355008}(14743,\cdot)\)
\(\chi_{355008}(16519,\cdot)\)
\(\chi_{355008}(16807,\cdot)\)
\(\chi_{355008}(18583,\cdot)\)
\(\chi_{355008}(18871,\cdot)\)
\(\chi_{355008}(20647,\cdot)\)
\(\chi_{355008}(20935,\cdot)\)
\(\chi_{355008}(22711,\cdot)\)
\(\chi_{355008}(22999,\cdot)\)
\(\chi_{355008}(24775,\cdot)\)
\(\chi_{355008}(25063,\cdot)\)
\(\chi_{355008}(26839,\cdot)\)
\(\chi_{355008}(27127,\cdot)\)
\(\chi_{355008}(28903,\cdot)\)
\(\chi_{355008}(29191,\cdot)\)
...
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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: |
$\Q(\zeta_{1032})$ |
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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 1032 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((321727,66565,118337,85057)\) → \((-1,e\left(\frac{3}{8}\right),1,e\left(\frac{85}{258}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 355008 }(2359, a) \) |
\(1\) | \(1\) | \(e\left(\frac{991}{1032}\right)\) | \(e\left(\frac{499}{516}\right)\) | \(e\left(\frac{249}{344}\right)\) | \(e\left(\frac{269}{1032}\right)\) | \(e\left(\frac{149}{258}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{95}{516}\right)\) | \(e\left(\frac{475}{516}\right)\) | \(e\left(\frac{653}{1032}\right)\) | \(e\left(\frac{223}{258}\right)\) |
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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)
