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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(52020, base_ring=CyclotomicField(136))
M = H._module
chi = DirichletCharacter(H, M([68,0,34,101]))
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gp:[g,chi] = znchar(Mod(12367, 52020))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("52020.12367");
| Modulus: | \(52020\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(5780\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(136\) |
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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 \(\chi_{5780}(807,\cdot)\) |
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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: | 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);
|
\(\chi_{52020}(127,\cdot)\)
\(\chi_{52020}(1243,\cdot)\)
\(\chi_{52020}(1783,\cdot)\)
\(\chi_{52020}(2287,\cdot)\)
\(\chi_{52020}(3187,\cdot)\)
\(\chi_{52020}(4303,\cdot)\)
\(\chi_{52020}(4843,\cdot)\)
\(\chi_{52020}(5347,\cdot)\)
\(\chi_{52020}(6247,\cdot)\)
\(\chi_{52020}(7363,\cdot)\)
\(\chi_{52020}(7903,\cdot)\)
\(\chi_{52020}(8407,\cdot)\)
\(\chi_{52020}(9307,\cdot)\)
\(\chi_{52020}(10423,\cdot)\)
\(\chi_{52020}(10963,\cdot)\)
\(\chi_{52020}(11467,\cdot)\)
\(\chi_{52020}(12367,\cdot)\)
\(\chi_{52020}(13483,\cdot)\)
\(\chi_{52020}(14023,\cdot)\)
\(\chi_{52020}(14527,\cdot)\)
\(\chi_{52020}(16543,\cdot)\)
\(\chi_{52020}(17083,\cdot)\)
\(\chi_{52020}(17587,\cdot)\)
\(\chi_{52020}(18487,\cdot)\)
\(\chi_{52020}(19603,\cdot)\)
\(\chi_{52020}(20143,\cdot)\)
\(\chi_{52020}(20647,\cdot)\)
\(\chi_{52020}(21547,\cdot)\)
\(\chi_{52020}(22663,\cdot)\)
\(\chi_{52020}(23203,\cdot)\)
...
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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 };
\((26011,28901,41617,41041)\) → \((-1,1,i,e\left(\frac{101}{136}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 52020 }(12367, a) \) |
\(1\) | \(1\) | \(e\left(\frac{117}{136}\right)\) | \(e\left(\frac{79}{136}\right)\) | \(e\left(\frac{21}{68}\right)\) | \(e\left(\frac{27}{68}\right)\) | \(e\left(\frac{117}{136}\right)\) | \(e\left(\frac{45}{136}\right)\) | \(e\left(\frac{25}{136}\right)\) | \(e\left(\frac{7}{136}\right)\) | \(e\left(\frac{135}{136}\right)\) | \(e\left(\frac{3}{34}\right)\) |
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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)
