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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(11005, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([105,154,18]))
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gp:[g,chi] = znchar(Mod(1564, 11005))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("11005.1564");
| Modulus: | \(11005\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(11005\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(210\) |
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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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\(\chi_{11005}(299,\cdot)\)
\(\chi_{11005}(324,\cdot)\)
\(\chi_{11005}(379,\cdot)\)
\(\chi_{11005}(524,\cdot)\)
\(\chi_{11005}(784,\cdot)\)
\(\chi_{11005}(824,\cdot)\)
\(\chi_{11005}(1144,\cdot)\)
\(\chi_{11005}(1409,\cdot)\)
\(\chi_{11005}(1529,\cdot)\)
\(\chi_{11005}(1564,\cdot)\)
\(\chi_{11005}(1994,\cdot)\)
\(\chi_{11005}(2004,\cdot)\)
\(\chi_{11005}(2219,\cdot)\)
\(\chi_{11005}(2779,\cdot)\)
\(\chi_{11005}(2994,\cdot)\)
\(\chi_{11005}(3089,\cdot)\)
\(\chi_{11005}(3269,\cdot)\)
\(\chi_{11005}(3324,\cdot)\)
\(\chi_{11005}(3934,\cdot)\)
\(\chi_{11005}(4639,\cdot)\)
\(\chi_{11005}(4979,\cdot)\)
\(\chi_{11005}(5744,\cdot)\)
\(\chi_{11005}(5794,\cdot)\)
\(\chi_{11005}(6054,\cdot)\)
\(\chi_{11005}(6559,\cdot)\)
\(\chi_{11005}(7044,\cdot)\)
\(\chi_{11005}(7109,\cdot)\)
\(\chi_{11005}(7149,\cdot)\)
\(\chi_{11005}(7179,\cdot)\)
\(\chi_{11005}(7189,\cdot)\)
...
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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 };
\((2202,3196,10231)\) → \((-1,e\left(\frac{11}{15}\right),e\left(\frac{3}{35}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 11005 }(1564, a) \) |
\(1\) | \(1\) | \(e\left(\frac{43}{70}\right)\) | \(e\left(\frac{97}{210}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{8}{105}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{59}{70}\right)\) | \(e\left(\frac{97}{105}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{191}{210}\right)\) |
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sage:chi(x)
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gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
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magma:chi(x)
