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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8536, base_ring=CyclotomicField(240))
M = H._module
chi = DirichletCharacter(H, M([120,120,48,35]))
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gp:[g,chi] = znchar(Mod(1115, 8536))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8536.1115");
| Modulus: | \(8536\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(8536\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(240\) |
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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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\(\chi_{8536}(3,\cdot)\)
\(\chi_{8536}(163,\cdot)\)
\(\chi_{8536}(323,\cdot)\)
\(\chi_{8536}(339,\cdot)\)
\(\chi_{8536}(779,\cdot)\)
\(\chi_{8536}(939,\cdot)\)
\(\chi_{8536}(995,\cdot)\)
\(\chi_{8536}(1115,\cdot)\)
\(\chi_{8536}(1259,\cdot)\)
\(\chi_{8536}(1347,\cdot)\)
\(\chi_{8536}(1411,\cdot)\)
\(\chi_{8536}(1499,\cdot)\)
\(\chi_{8536}(1555,\cdot)\)
\(\chi_{8536}(1875,\cdot)\)
\(\chi_{8536}(2187,\cdot)\)
\(\chi_{8536}(2275,\cdot)\)
\(\chi_{8536}(2491,\cdot)\)
\(\chi_{8536}(2667,\cdot)\)
\(\chi_{8536}(2907,\cdot)\)
\(\chi_{8536}(2963,\cdot)\)
\(\chi_{8536}(3051,\cdot)\)
\(\chi_{8536}(3107,\cdot)\)
\(\chi_{8536}(3347,\cdot)\)
\(\chi_{8536}(3523,\cdot)\)
\(\chi_{8536}(3683,\cdot)\)
\(\chi_{8536}(4123,\cdot)\)
\(\chi_{8536}(4139,\cdot)\)
\(\chi_{8536}(4299,\cdot)\)
\(\chi_{8536}(4459,\cdot)\)
\(\chi_{8536}(4515,\cdot)\)
...
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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 };
\((2135,4269,1553,6601)\) → \((-1,-1,e\left(\frac{1}{5}\right),e\left(\frac{7}{48}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 8536 }(1115, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{97}{120}\right)\) | \(e\left(\frac{107}{240}\right)\) | \(e\left(\frac{101}{240}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{83}{240}\right)\) | \(e\left(\frac{61}{240}\right)\) | \(e\left(\frac{187}{240}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{35}{48}\right)\) |
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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)
