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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([105,14,99]))
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gp:[g,chi] = znchar(Mod(1249, 11005))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("11005.1249");
| 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: | 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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\(\chi_{11005}(479,\cdot)\)
\(\chi_{11005}(599,\cdot)\)
\(\chi_{11005}(919,\cdot)\)
\(\chi_{11005}(979,\cdot)\)
\(\chi_{11005}(1229,\cdot)\)
\(\chi_{11005}(1249,\cdot)\)
\(\chi_{11005}(1259,\cdot)\)
\(\chi_{11005}(1569,\cdot)\)
\(\chi_{11005}(1609,\cdot)\)
\(\chi_{11005}(1874,\cdot)\)
\(\chi_{11005}(3159,\cdot)\)
\(\chi_{11005}(3389,\cdot)\)
\(\chi_{11005}(3399,\cdot)\)
\(\chi_{11005}(3469,\cdot)\)
\(\chi_{11005}(3699,\cdot)\)
\(\chi_{11005}(3734,\cdot)\)
\(\chi_{11005}(4039,\cdot)\)
\(\chi_{11005}(4244,\cdot)\)
\(\chi_{11005}(4254,\cdot)\)
\(\chi_{11005}(4659,\cdot)\)
\(\chi_{11005}(4699,\cdot)\)
\(\chi_{11005}(4719,\cdot)\)
\(\chi_{11005}(5949,\cdot)\)
\(\chi_{11005}(6269,\cdot)\)
\(\chi_{11005}(6524,\cdot)\)
\(\chi_{11005}(6529,\cdot)\)
\(\chi_{11005}(7144,\cdot)\)
\(\chi_{11005}(7644,\cdot)\)
\(\chi_{11005}(7664,\cdot)\)
\(\chi_{11005}(7974,\cdot)\)
...
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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 };
\((2202,3196,10231)\) → \((-1,e\left(\frac{1}{15}\right),e\left(\frac{33}{70}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 11005 }(1249, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{173}{210}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{79}{105}\right)\) | \(e\left(\frac{88}{105}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{68}{105}\right)\) | \(e\left(\frac{31}{210}\right)\) | \(e\left(\frac{143}{210}\right)\) | \(e\left(\frac{13}{21}\right)\) |
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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)
