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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10580, base_ring=CyclotomicField(1012))
M = H._module
chi = DirichletCharacter(H, M([506,759,54]))
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gp:[g,chi] = znchar(Mod(43, 10580))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("10580.43");
| Modulus: | \(10580\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(10580\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1012\) |
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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);
|
\(\chi_{10580}(7,\cdot)\)
\(\chi_{10580}(43,\cdot)\)
\(\chi_{10580}(67,\cdot)\)
\(\chi_{10580}(83,\cdot)\)
\(\chi_{10580}(103,\cdot)\)
\(\chi_{10580}(107,\cdot)\)
\(\chi_{10580}(143,\cdot)\)
\(\chi_{10580}(203,\cdot)\)
\(\chi_{10580}(227,\cdot)\)
\(\chi_{10580}(247,\cdot)\)
\(\chi_{10580}(267,\cdot)\)
\(\chi_{10580}(283,\cdot)\)
\(\chi_{10580}(287,\cdot)\)
\(\chi_{10580}(327,\cdot)\)
\(\chi_{10580}(343,\cdot)\)
\(\chi_{10580}(383,\cdot)\)
\(\chi_{10580}(387,\cdot)\)
\(\chi_{10580}(447,\cdot)\)
\(\chi_{10580}(467,\cdot)\)
\(\chi_{10580}(503,\cdot)\)
\(\chi_{10580}(523,\cdot)\)
\(\chi_{10580}(527,\cdot)\)
\(\chi_{10580}(543,\cdot)\)
\(\chi_{10580}(563,\cdot)\)
\(\chi_{10580}(567,\cdot)\)
\(\chi_{10580}(603,\cdot)\)
\(\chi_{10580}(663,\cdot)\)
\(\chi_{10580}(687,\cdot)\)
\(\chi_{10580}(707,\cdot)\)
\(\chi_{10580}(723,\cdot)\)
...
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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 };
\((5291,2117,2121)\) → \((-1,-i,e\left(\frac{27}{506}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 10580 }(43, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{611}{1012}\right)\) | \(e\left(\frac{135}{1012}\right)\) | \(e\left(\frac{105}{506}\right)\) | \(e\left(\frac{116}{253}\right)\) | \(e\left(\frac{393}{1012}\right)\) | \(e\left(\frac{301}{1012}\right)\) | \(e\left(\frac{317}{506}\right)\) | \(e\left(\frac{373}{506}\right)\) | \(e\left(\frac{821}{1012}\right)\) | \(e\left(\frac{255}{506}\right)\) |
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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)
