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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(125341, base_ring=CyclotomicField(720))
M = H._module
chi = DirichletCharacter(H, M([225,700,108]))
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gp:[g,chi] = znchar(Mod(549, 125341))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("125341.549");
| Modulus: | \(125341\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(125341\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(720\) |
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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);
|
\(\chi_{125341}(549,\cdot)\)
\(\chi_{125341}(1673,\cdot)\)
\(\chi_{125341}(3907,\cdot)\)
\(\chi_{125341}(4100,\cdot)\)
\(\chi_{125341}(5208,\cdot)\)
\(\chi_{125341}(5688,\cdot)\)
\(\chi_{125341}(5700,\cdot)\)
\(\chi_{125341}(5998,\cdot)\)
\(\chi_{125341}(6735,\cdot)\)
\(\chi_{125341}(6824,\cdot)\)
\(\chi_{125341}(8748,\cdot)\)
\(\chi_{125341}(9058,\cdot)\)
\(\chi_{125341}(10341,\cdot)\)
\(\chi_{125341}(10343,\cdot)\)
\(\chi_{125341}(10839,\cdot)\)
\(\chi_{125341}(11776,\cdot)\)
\(\chi_{125341}(12568,\cdot)\)
\(\chi_{125341}(14200,\cdot)\)
\(\chi_{125341}(15295,\cdot)\)
\(\chi_{125341}(15616,\cdot)\)
\(\chi_{125341}(15926,\cdot)\)
\(\chi_{125341}(18926,\cdot)\)
\(\chi_{125341}(19351,\cdot)\)
\(\chi_{125341}(20244,\cdot)\)
\(\chi_{125341}(20269,\cdot)\)
\(\chi_{125341}(20446,\cdot)\)
\(\chi_{125341}(20767,\cdot)\)
\(\chi_{125341}(22757,\cdot)\)
\(\chi_{125341}(23994,\cdot)\)
\(\chi_{125341}(25087,\cdot)\)
...
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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 };
\((22120,46360,8688)\) → \((e\left(\frac{5}{16}\right),e\left(\frac{35}{36}\right),e\left(\frac{3}{20}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 125341 }(549, a) \) |
\(1\) | \(1\) | \(e\left(\frac{109}{360}\right)\) | \(e\left(\frac{119}{240}\right)\) | \(e\left(\frac{109}{180}\right)\) | \(e\left(\frac{97}{720}\right)\) | \(e\left(\frac{115}{144}\right)\) | \(e\left(\frac{209}{240}\right)\) | \(e\left(\frac{109}{120}\right)\) | \(e\left(\frac{119}{120}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{439}{720}\right)\) |
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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)
