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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(87451, base_ring=CyclotomicField(930))
M = H._module
chi = DirichletCharacter(H, M([310,775,277]))
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gp:[g,chi] = znchar(Mod(5938, 87451))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("87451.5938");
| Modulus: | \(87451\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(87451\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(930\) |
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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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\(\chi_{87451}(296,\cdot)\)
\(\chi_{87451}(641,\cdot)\)
\(\chi_{87451}(933,\cdot)\)
\(\chi_{87451}(1005,\cdot)\)
\(\chi_{87451}(1096,\cdot)\)
\(\chi_{87451}(1388,\cdot)\)
\(\chi_{87451}(1934,\cdot)\)
\(\chi_{87451}(2006,\cdot)\)
\(\chi_{87451}(3117,\cdot)\)
\(\chi_{87451}(3462,\cdot)\)
\(\chi_{87451}(3754,\cdot)\)
\(\chi_{87451}(3826,\cdot)\)
\(\chi_{87451}(3917,\cdot)\)
\(\chi_{87451}(4209,\cdot)\)
\(\chi_{87451}(4755,\cdot)\)
\(\chi_{87451}(4827,\cdot)\)
\(\chi_{87451}(5938,\cdot)\)
\(\chi_{87451}(6283,\cdot)\)
\(\chi_{87451}(6575,\cdot)\)
\(\chi_{87451}(6647,\cdot)\)
\(\chi_{87451}(6738,\cdot)\)
\(\chi_{87451}(7030,\cdot)\)
\(\chi_{87451}(7576,\cdot)\)
\(\chi_{87451}(7648,\cdot)\)
\(\chi_{87451}(8759,\cdot)\)
\(\chi_{87451}(9104,\cdot)\)
\(\chi_{87451}(9396,\cdot)\)
\(\chi_{87451}(9468,\cdot)\)
\(\chi_{87451}(9559,\cdot)\)
\(\chi_{87451}(9851,\cdot)\)
...
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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 };
\((74959,73998,39404)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{5}{6}\right),e\left(\frac{277}{930}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 87451 }(5938, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{111}{310}\right)\) | \(e\left(\frac{299}{310}\right)\) | \(e\left(\frac{111}{155}\right)\) | \(e\left(\frac{41}{186}\right)\) | \(e\left(\frac{10}{31}\right)\) | \(e\left(\frac{23}{310}\right)\) | \(e\left(\frac{144}{155}\right)\) | \(e\left(\frac{269}{465}\right)\) | \(e\left(\frac{173}{465}\right)\) | \(e\left(\frac{211}{310}\right)\) |
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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)
