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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9595, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([135,110,153]))
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gp:[g,chi] = znchar(Mod(4708, 9595))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9595.4708");
| Modulus: | \(9595\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(9595\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(180\) |
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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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\(\chi_{9595}(243,\cdot)\)
\(\chi_{9595}(262,\cdot)\)
\(\chi_{9595}(668,\cdot)\)
\(\chi_{9595}(877,\cdot)\)
\(\chi_{9595}(1143,\cdot)\)
\(\chi_{9595}(1173,\cdot)\)
\(\chi_{9595}(1352,\cdot)\)
\(\chi_{9595}(1382,\cdot)\)
\(\chi_{9595}(1648,\cdot)\)
\(\chi_{9595}(1758,\cdot)\)
\(\chi_{9595}(1777,\cdot)\)
\(\chi_{9595}(1857,\cdot)\)
\(\chi_{9595}(2263,\cdot)\)
\(\chi_{9595}(2282,\cdot)\)
\(\chi_{9595}(2768,\cdot)\)
\(\chi_{9595}(2787,\cdot)\)
\(\chi_{9595}(2872,\cdot)\)
\(\chi_{9595}(3188,\cdot)\)
\(\chi_{9595}(3377,\cdot)\)
\(\chi_{9595}(4703,\cdot)\)
\(\chi_{9595}(4708,\cdot)\)
\(\chi_{9595}(4917,\cdot)\)
\(\chi_{9595}(5183,\cdot)\)
\(\chi_{9595}(5208,\cdot)\)
\(\chi_{9595}(5392,\cdot)\)
\(\chi_{9595}(5713,\cdot)\)
\(\chi_{9595}(5798,\cdot)\)
\(\chi_{9595}(5817,\cdot)\)
\(\chi_{9595}(6223,\cdot)\)
\(\chi_{9595}(6303,\cdot)\)
...
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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 };
\((7677,3536,7981)\) → \((-i,e\left(\frac{11}{18}\right),e\left(\frac{17}{20}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 9595 }(4708, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{19}{90}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{73}{180}\right)\) |
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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)
