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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4513, base_ring=CyclotomicField(564))
M = H._module
chi = DirichletCharacter(H, M([533]))
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gp:[g,chi] = znchar(Mod(88, 4513))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4513.88");
| Modulus: | \(4513\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4513\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(564\) |
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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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\(\chi_{4513}(11,\cdot)\)
\(\chi_{4513}(22,\cdot)\)
\(\chi_{4513}(37,\cdot)\)
\(\chi_{4513}(44,\cdot)\)
\(\chi_{4513}(61,\cdot)\)
\(\chi_{4513}(74,\cdot)\)
\(\chi_{4513}(88,\cdot)\)
\(\chi_{4513}(107,\cdot)\)
\(\chi_{4513}(122,\cdot)\)
\(\chi_{4513}(148,\cdot)\)
\(\chi_{4513}(157,\cdot)\)
\(\chi_{4513}(176,\cdot)\)
\(\chi_{4513}(214,\cdot)\)
\(\chi_{4513}(223,\cdot)\)
\(\chi_{4513}(244,\cdot)\)
\(\chi_{4513}(296,\cdot)\)
\(\chi_{4513}(314,\cdot)\)
\(\chi_{4513}(352,\cdot)\)
\(\chi_{4513}(359,\cdot)\)
\(\chi_{4513}(379,\cdot)\)
\(\chi_{4513}(387,\cdot)\)
\(\chi_{4513}(411,\cdot)\)
\(\chi_{4513}(425,\cdot)\)
\(\chi_{4513}(428,\cdot)\)
\(\chi_{4513}(446,\cdot)\)
\(\chi_{4513}(488,\cdot)\)
\(\chi_{4513}(511,\cdot)\)
\(\chi_{4513}(592,\cdot)\)
\(\chi_{4513}(628,\cdot)\)
\(\chi_{4513}(717,\cdot)\)
...
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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 };
\(7\) → \(e\left(\frac{533}{564}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 4513 }(88, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{47}\right)\) | \(e\left(\frac{21}{47}\right)\) | \(e\left(\frac{22}{47}\right)\) | \(e\left(\frac{149}{188}\right)\) | \(e\left(\frac{32}{47}\right)\) | \(e\left(\frac{533}{564}\right)\) | \(e\left(\frac{33}{47}\right)\) | \(e\left(\frac{42}{47}\right)\) | \(e\left(\frac{5}{188}\right)\) | \(e\left(\frac{131}{141}\right)\) |
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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)
