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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4150, base_ring=CyclotomicField(82))
M = H._module
chi = DirichletCharacter(H, M([41,78]))
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gp:[g,chi] = znchar(Mod(2599, 4150))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4150.2599");
| Modulus: | \(4150\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(415\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(82\) |
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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 \(\chi_{415}(109,\cdot)\) |
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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: | even |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{4150}(49,\cdot)\)
\(\chi_{4150}(99,\cdot)\)
\(\chi_{4150}(199,\cdot)\)
\(\chi_{4150}(349,\cdot)\)
\(\chi_{4150}(549,\cdot)\)
\(\chi_{4150}(649,\cdot)\)
\(\chi_{4150}(899,\cdot)\)
\(\chi_{4150}(949,\cdot)\)
\(\chi_{4150}(999,\cdot)\)
\(\chi_{4150}(1149,\cdot)\)
\(\chi_{4150}(1199,\cdot)\)
\(\chi_{4150}(1249,\cdot)\)
\(\chi_{4150}(1349,\cdot)\)
\(\chi_{4150}(1449,\cdot)\)
\(\chi_{4150}(1849,\cdot)\)
\(\chi_{4150}(1949,\cdot)\)
\(\chi_{4150}(1999,\cdot)\)
\(\chi_{4150}(2199,\cdot)\)
\(\chi_{4150}(2349,\cdot)\)
\(\chi_{4150}(2399,\cdot)\)
\(\chi_{4150}(2499,\cdot)\)
\(\chi_{4150}(2549,\cdot)\)
\(\chi_{4150}(2599,\cdot)\)
\(\chi_{4150}(2749,\cdot)\)
\(\chi_{4150}(2849,\cdot)\)
\(\chi_{4150}(2899,\cdot)\)
\(\chi_{4150}(2949,\cdot)\)
\(\chi_{4150}(2999,\cdot)\)
\(\chi_{4150}(3049,\cdot)\)
\(\chi_{4150}(3099,\cdot)\)
...
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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 };
\((1827,251)\) → \((-1,e\left(\frac{39}{41}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 4150 }(2599, a) \) |
\(1\) | \(1\) | \(e\left(\frac{81}{82}\right)\) | \(e\left(\frac{9}{82}\right)\) | \(e\left(\frac{40}{41}\right)\) | \(e\left(\frac{34}{41}\right)\) | \(e\left(\frac{61}{82}\right)\) | \(e\left(\frac{63}{82}\right)\) | \(e\left(\frac{29}{41}\right)\) | \(e\left(\frac{4}{41}\right)\) | \(e\left(\frac{47}{82}\right)\) | \(e\left(\frac{79}{82}\right)\) |
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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)
