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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(54691, base_ring=CyclotomicField(150))
M = H._module
chi = DirichletCharacter(H, M([75,125,67]))
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gp:[g,chi] = znchar(Mod(1791, 54691))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("54691.1791");
| Modulus: | \(54691\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(54691\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(150\) |
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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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\(\chi_{54691}(1154,\cdot)\)
\(\chi_{54691}(1252,\cdot)\)
\(\chi_{54691}(1427,\cdot)\)
\(\chi_{54691}(1791,\cdot)\)
\(\chi_{54691}(2337,\cdot)\)
\(\chi_{54691}(3338,\cdot)\)
\(\chi_{54691}(4437,\cdot)\)
\(\chi_{54691}(7342,\cdot)\)
\(\chi_{54691}(7531,\cdot)\)
\(\chi_{54691}(7804,\cdot)\)
\(\chi_{54691}(8252,\cdot)\)
\(\chi_{54691}(13439,\cdot)\)
\(\chi_{54691}(14720,\cdot)\)
\(\chi_{54691}(15532,\cdot)\)
\(\chi_{54691}(15623,\cdot)\)
\(\chi_{54691}(16540,\cdot)\)
\(\chi_{54691}(18178,\cdot)\)
\(\chi_{54691}(18535,\cdot)\)
\(\chi_{54691}(19088,\cdot)\)
\(\chi_{54691}(19907,\cdot)\)
\(\chi_{54691}(20362,\cdot)\)
\(\chi_{54691}(20901,\cdot)\)
\(\chi_{54691}(20999,\cdot)\)
\(\chi_{54691}(22637,\cdot)\)
\(\chi_{54691}(23358,\cdot)\)
\(\chi_{54691}(25640,\cdot)\)
\(\chi_{54691}(26998,\cdot)\)
\(\chi_{54691}(29644,\cdot)\)
\(\chi_{54691}(30911,\cdot)\)
\(\chi_{54691}(35559,\cdot)\)
...
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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 };
\((15627,21036,45683)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{67}{150}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 54691 }(1791, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{119}{150}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{44}{75}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{31}{75}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{31}{150}\right)\) |
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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)
