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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6035, base_ring=CyclotomicField(560))
M = H._module
chi = DirichletCharacter(H, M([140,35,256]))
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gp:[g,chi] = znchar(Mod(1142, 6035))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6035.1142");
| Modulus: | \(6035\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(6035\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(560\) |
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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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\(\chi_{6035}(12,\cdot)\)
\(\chi_{6035}(58,\cdot)\)
\(\chi_{6035}(107,\cdot)\)
\(\chi_{6035}(182,\cdot)\)
\(\chi_{6035}(192,\cdot)\)
\(\chi_{6035}(228,\cdot)\)
\(\chi_{6035}(277,\cdot)\)
\(\chi_{6035}(292,\cdot)\)
\(\chi_{6035}(313,\cdot)\)
\(\chi_{6035}(333,\cdot)\)
\(\chi_{6035}(363,\cdot)\)
\(\chi_{6035}(398,\cdot)\)
\(\chi_{6035}(453,\cdot)\)
\(\chi_{6035}(462,\cdot)\)
\(\chi_{6035}(503,\cdot)\)
\(\chi_{6035}(507,\cdot)\)
\(\chi_{6035}(533,\cdot)\)
\(\chi_{6035}(547,\cdot)\)
\(\chi_{6035}(592,\cdot)\)
\(\chi_{6035}(617,\cdot)\)
\(\chi_{6035}(618,\cdot)\)
\(\chi_{6035}(632,\cdot)\)
\(\chi_{6035}(677,\cdot)\)
\(\chi_{6035}(703,\cdot)\)
\(\chi_{6035}(787,\cdot)\)
\(\chi_{6035}(793,\cdot)\)
\(\chi_{6035}(862,\cdot)\)
\(\chi_{6035}(932,\cdot)\)
\(\chi_{6035}(947,\cdot)\)
\(\chi_{6035}(963,\cdot)\)
...
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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 };
\((3622,5681,3486)\) → \((i,e\left(\frac{1}{16}\right),e\left(\frac{16}{35}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 6035 }(1142, a) \) |
\(1\) | \(1\) | \(e\left(\frac{243}{280}\right)\) | \(e\left(\frac{391}{560}\right)\) | \(e\left(\frac{103}{140}\right)\) | \(e\left(\frac{317}{560}\right)\) | \(e\left(\frac{221}{560}\right)\) | \(e\left(\frac{169}{280}\right)\) | \(e\left(\frac{111}{280}\right)\) | \(e\left(\frac{341}{560}\right)\) | \(e\left(\frac{243}{560}\right)\) | \(e\left(\frac{29}{35}\right)\) |
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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)
