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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5529, base_ring=CyclotomicField(144))
M = H._module
chi = DirichletCharacter(H, M([72,8,117]))
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gp:[g,chi] = znchar(Mod(1085, 5529))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5529.1085");
| Modulus: | \(5529\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(5529\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(144\) |
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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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\(\chi_{5529}(89,\cdot)\)
\(\chi_{5529}(167,\cdot)\)
\(\chi_{5529}(212,\cdot)\)
\(\chi_{5529}(299,\cdot)\)
\(\chi_{5529}(458,\cdot)\)
\(\chi_{5529}(497,\cdot)\)
\(\chi_{5529}(794,\cdot)\)
\(\chi_{5529}(1040,\cdot)\)
\(\chi_{5529}(1055,\cdot)\)
\(\chi_{5529}(1079,\cdot)\)
\(\chi_{5529}(1085,\cdot)\)
\(\chi_{5529}(1172,\cdot)\)
\(\chi_{5529}(1340,\cdot)\)
\(\chi_{5529}(1370,\cdot)\)
\(\chi_{5529}(1637,\cdot)\)
\(\chi_{5529}(1667,\cdot)\)
\(\chi_{5529}(1913,\cdot)\)
\(\chi_{5529}(1922,\cdot)\)
\(\chi_{5529}(1952,\cdot)\)
\(\chi_{5529}(1967,\cdot)\)
\(\chi_{5529}(2219,\cdot)\)
\(\chi_{5529}(2504,\cdot)\)
\(\chi_{5529}(2510,\cdot)\)
\(\chi_{5529}(2540,\cdot)\)
\(\chi_{5529}(2549,\cdot)\)
\(\chi_{5529}(2708,\cdot)\)
\(\chi_{5529}(2795,\cdot)\)
\(\chi_{5529}(2825,\cdot)\)
\(\chi_{5529}(3092,\cdot)\)
\(\chi_{5529}(3131,\cdot)\)
...
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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 };
\((1844,4657,1654)\) → \((-1,e\left(\frac{1}{18}\right),e\left(\frac{13}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 5529 }(1085, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{72}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{29}{144}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{55}{144}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{85}{144}\right)\) | \(e\left(\frac{101}{144}\right)\) | \(e\left(\frac{13}{18}\right)\) |
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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)
