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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8775, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([50,171,30]))
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gp:[g,chi] = znchar(Mod(3488, 8775))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8775.3488");
| Modulus: | \(8775\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(8775\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(180\) |
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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: | 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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\(\chi_{8775}(212,\cdot)\)
\(\chi_{8775}(452,\cdot)\)
\(\chi_{8775}(563,\cdot)\)
\(\chi_{8775}(797,\cdot)\)
\(\chi_{8775}(803,\cdot)\)
\(\chi_{8775}(1037,\cdot)\)
\(\chi_{8775}(1148,\cdot)\)
\(\chi_{8775}(1388,\cdot)\)
\(\chi_{8775}(1622,\cdot)\)
\(\chi_{8775}(1733,\cdot)\)
\(\chi_{8775}(1967,\cdot)\)
\(\chi_{8775}(1973,\cdot)\)
\(\chi_{8775}(2552,\cdot)\)
\(\chi_{8775}(2558,\cdot)\)
\(\chi_{8775}(2792,\cdot)\)
\(\chi_{8775}(2903,\cdot)\)
\(\chi_{8775}(3137,\cdot)\)
\(\chi_{8775}(3377,\cdot)\)
\(\chi_{8775}(3488,\cdot)\)
\(\chi_{8775}(3722,\cdot)\)
\(\chi_{8775}(3728,\cdot)\)
\(\chi_{8775}(3962,\cdot)\)
\(\chi_{8775}(4073,\cdot)\)
\(\chi_{8775}(4313,\cdot)\)
\(\chi_{8775}(4547,\cdot)\)
\(\chi_{8775}(4658,\cdot)\)
\(\chi_{8775}(4892,\cdot)\)
\(\chi_{8775}(4898,\cdot)\)
\(\chi_{8775}(5477,\cdot)\)
\(\chi_{8775}(5483,\cdot)\)
...
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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: |
$\Q(\zeta_{180})$ |
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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 180 polynomial (not computed) |
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sage:chi.fixed_field()
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\((326,352,8101)\) → \((e\left(\frac{5}{18}\right),e\left(\frac{19}{20}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
| \( \chi_{ 8775 }(3488, a) \) |
\(1\) | \(1\) | \(e\left(\frac{71}{180}\right)\) | \(e\left(\frac{71}{90}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{67}{180}\right)\) |
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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)
