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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(22599, base_ring=CyclotomicField(810))
M = H._module
chi = DirichletCharacter(H, M([685,486]))
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gp:[g,chi] = znchar(Mod(1151, 22599))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("22599.1151");
| Modulus: | \(22599\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(7533\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(810\) |
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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: | no, induced from \(\chi_{7533}(1616,\cdot)\) |
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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: | no |
| 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);
|
\(\chi_{22599}(8,\cdot)\)
\(\chi_{22599}(35,\cdot)\)
\(\chi_{22599}(233,\cdot)\)
\(\chi_{22599}(287,\cdot)\)
\(\chi_{22599}(314,\cdot)\)
\(\chi_{22599}(467,\cdot)\)
\(\chi_{22599}(746,\cdot)\)
\(\chi_{22599}(791,\cdot)\)
\(\chi_{22599}(845,\cdot)\)
\(\chi_{22599}(872,\cdot)\)
\(\chi_{22599}(1070,\cdot)\)
\(\chi_{22599}(1124,\cdot)\)
\(\chi_{22599}(1151,\cdot)\)
\(\chi_{22599}(1304,\cdot)\)
\(\chi_{22599}(1583,\cdot)\)
\(\chi_{22599}(1628,\cdot)\)
\(\chi_{22599}(1682,\cdot)\)
\(\chi_{22599}(1709,\cdot)\)
\(\chi_{22599}(1907,\cdot)\)
\(\chi_{22599}(1961,\cdot)\)
\(\chi_{22599}(1988,\cdot)\)
\(\chi_{22599}(2141,\cdot)\)
\(\chi_{22599}(2420,\cdot)\)
\(\chi_{22599}(2465,\cdot)\)
\(\chi_{22599}(2519,\cdot)\)
\(\chi_{22599}(2546,\cdot)\)
\(\chi_{22599}(2744,\cdot)\)
\(\chi_{22599}(2798,\cdot)\)
\(\chi_{22599}(2825,\cdot)\)
\(\chi_{22599}(2978,\cdot)\)
...
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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 };
\((21143,2917)\) → \((e\left(\frac{137}{162}\right),e\left(\frac{3}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 22599 }(1151, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{199}{810}\right)\) | \(e\left(\frac{199}{405}\right)\) | \(e\left(\frac{73}{162}\right)\) | \(e\left(\frac{404}{405}\right)\) | \(e\left(\frac{199}{270}\right)\) | \(e\left(\frac{94}{135}\right)\) | \(e\left(\frac{103}{810}\right)\) | \(e\left(\frac{148}{405}\right)\) | \(e\left(\frac{197}{810}\right)\) | \(e\left(\frac{398}{405}\right)\) |
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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)
