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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(15200, base_ring=CyclotomicField(360))
M = H._module
chi = DirichletCharacter(H, M([180,225,324,100]))
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gp:[g,chi] = znchar(Mod(1419, 15200))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("15200.1419");
| Modulus: | \(15200\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(15200\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(360\) |
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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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\(\chi_{15200}(59,\cdot)\)
\(\chi_{15200}(219,\cdot)\)
\(\chi_{15200}(459,\cdot)\)
\(\chi_{15200}(659,\cdot)\)
\(\chi_{15200}(819,\cdot)\)
\(\chi_{15200}(979,\cdot)\)
\(\chi_{15200}(1059,\cdot)\)
\(\chi_{15200}(1219,\cdot)\)
\(\chi_{15200}(1419,\cdot)\)
\(\chi_{15200}(1459,\cdot)\)
\(\chi_{15200}(1579,\cdot)\)
\(\chi_{15200}(1739,\cdot)\)
\(\chi_{15200}(1819,\cdot)\)
\(\chi_{15200}(1979,\cdot)\)
\(\chi_{15200}(2179,\cdot)\)
\(\chi_{15200}(2219,\cdot)\)
\(\chi_{15200}(2339,\cdot)\)
\(\chi_{15200}(2579,\cdot)\)
\(\chi_{15200}(2739,\cdot)\)
\(\chi_{15200}(2939,\cdot)\)
\(\chi_{15200}(2979,\cdot)\)
\(\chi_{15200}(3259,\cdot)\)
\(\chi_{15200}(3339,\cdot)\)
\(\chi_{15200}(3739,\cdot)\)
\(\chi_{15200}(3859,\cdot)\)
\(\chi_{15200}(4019,\cdot)\)
\(\chi_{15200}(4259,\cdot)\)
\(\chi_{15200}(4459,\cdot)\)
\(\chi_{15200}(4619,\cdot)\)
\(\chi_{15200}(4779,\cdot)\)
...
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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 };
\((12351,5701,13377,8001)\) → \((-1,e\left(\frac{5}{8}\right),e\left(\frac{9}{10}\right),e\left(\frac{5}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 15200 }(1419, a) \) |
\(1\) | \(1\) | \(e\left(\frac{103}{360}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{103}{180}\right)\) | \(e\left(\frac{43}{120}\right)\) | \(e\left(\frac{311}{360}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{73}{360}\right)\) | \(e\left(\frac{127}{180}\right)\) | \(e\left(\frac{103}{120}\right)\) | \(e\left(\frac{143}{360}\right)\) |
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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)
