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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7448, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([0,0,3,112]))
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gp:[g,chi] = znchar(Mod(689, 7448))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7448.689");
| Modulus: | \(7448\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(931\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(126\) |
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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 \(\chi_{931}(689,\cdot)\) |
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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: | odd |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{7448}(17,\cdot)\)
\(\chi_{7448}(73,\cdot)\)
\(\chi_{7448}(481,\cdot)\)
\(\chi_{7448}(593,\cdot)\)
\(\chi_{7448}(689,\cdot)\)
\(\chi_{7448}(1081,\cdot)\)
\(\chi_{7448}(1137,\cdot)\)
\(\chi_{7448}(1377,\cdot)\)
\(\chi_{7448}(1545,\cdot)\)
\(\chi_{7448}(1657,\cdot)\)
\(\chi_{7448}(1753,\cdot)\)
\(\chi_{7448}(2145,\cdot)\)
\(\chi_{7448}(2201,\cdot)\)
\(\chi_{7448}(2441,\cdot)\)
\(\chi_{7448}(2609,\cdot)\)
\(\chi_{7448}(2721,\cdot)\)
\(\chi_{7448}(2817,\cdot)\)
\(\chi_{7448}(3209,\cdot)\)
\(\chi_{7448}(3505,\cdot)\)
\(\chi_{7448}(3673,\cdot)\)
\(\chi_{7448}(3785,\cdot)\)
\(\chi_{7448}(3881,\cdot)\)
\(\chi_{7448}(4273,\cdot)\)
\(\chi_{7448}(4329,\cdot)\)
\(\chi_{7448}(4569,\cdot)\)
\(\chi_{7448}(4737,\cdot)\)
\(\chi_{7448}(4849,\cdot)\)
\(\chi_{7448}(4945,\cdot)\)
\(\chi_{7448}(5337,\cdot)\)
\(\chi_{7448}(5393,\cdot)\)
...
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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 };
\((1863,3725,3041,3137)\) → \((1,1,e\left(\frac{1}{42}\right),e\left(\frac{8}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 7448 }(689, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{73}{126}\right)\) | \(e\left(\frac{115}{126}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{29}{126}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{61}{126}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{31}{42}\right)\) |
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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)
