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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(28313, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([35,33]))
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gp:[g,chi] = znchar(Mod(132, 28313))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("28313.132");
| Modulus: | \(28313\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(28313\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(110\) |
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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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\(\chi_{28313}(132,\cdot)\)
\(\chi_{28313}(2061,\cdot)\)
\(\chi_{28313}(3503,\cdot)\)
\(\chi_{28313}(3825,\cdot)\)
\(\chi_{28313}(4523,\cdot)\)
\(\chi_{28313}(4734,\cdot)\)
\(\chi_{28313}(5056,\cdot)\)
\(\chi_{28313}(6615,\cdot)\)
\(\chi_{28313}(7196,\cdot)\)
\(\chi_{28313}(7518,\cdot)\)
\(\chi_{28313}(8216,\cdot)\)
\(\chi_{28313}(9077,\cdot)\)
\(\chi_{28313}(9447,\cdot)\)
\(\chi_{28313}(9658,\cdot)\)
\(\chi_{28313}(9980,\cdot)\)
\(\chi_{28313}(10889,\cdot)\)
\(\chi_{28313}(11211,\cdot)\)
\(\chi_{28313}(12770,\cdot)\)
\(\chi_{28313}(13140,\cdot)\)
\(\chi_{28313}(13351,\cdot)\)
\(\chi_{28313}(13673,\cdot)\)
\(\chi_{28313}(14001,\cdot)\)
\(\chi_{28313}(14371,\cdot)\)
\(\chi_{28313}(16833,\cdot)\)
\(\chi_{28313}(17694,\cdot)\)
\(\chi_{28313}(18925,\cdot)\)
\(\chi_{28313}(19295,\cdot)\)
\(\chi_{28313}(20526,\cdot)\)
\(\chi_{28313}(20737,\cdot)\)
\(\chi_{28313}(21059,\cdot)\)
...
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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 };
\((9849,23392)\) → \((e\left(\frac{7}{22}\right),e\left(\frac{3}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 28313 }(132, a) \) |
\(1\) | \(1\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{43}{110}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{27}{110}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{19}{22}\right)\) |
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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)
