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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4477, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([54,155]))
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gp:[g,chi] = znchar(Mod(2538, 4477))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4477.2538");
| Modulus: | \(4477\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(407\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(180\) |
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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 \(\chi_{407}(96,\cdot)\) |
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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);
|
\(\chi_{4477}(94,\cdot)\)
\(\chi_{4477}(161,\cdot)\)
\(\chi_{4477}(239,\cdot)\)
\(\chi_{4477}(457,\cdot)\)
\(\chi_{4477}(723,\cdot)\)
\(\chi_{4477}(838,\cdot)\)
\(\chi_{4477}(1086,\cdot)\)
\(\chi_{4477}(1129,\cdot)\)
\(\chi_{4477}(1201,\cdot)\)
\(\chi_{4477}(1371,\cdot)\)
\(\chi_{4477}(1425,\cdot)\)
\(\chi_{4477}(1613,\cdot)\)
\(\chi_{4477}(1667,\cdot)\)
\(\chi_{4477}(1685,\cdot)\)
\(\chi_{4477}(1734,\cdot)\)
\(\chi_{4477}(1855,\cdot)\)
\(\chi_{4477}(1909,\cdot)\)
\(\chi_{4477}(1976,\cdot)\)
\(\chi_{4477}(2030,\cdot)\)
\(\chi_{4477}(2048,\cdot)\)
\(\chi_{4477}(2054,\cdot)\)
\(\chi_{4477}(2151,\cdot)\)
\(\chi_{4477}(2218,\cdot)\)
\(\chi_{4477}(2272,\cdot)\)
\(\chi_{4477}(2296,\cdot)\)
\(\chi_{4477}(2460,\cdot)\)
\(\chi_{4477}(2514,\cdot)\)
\(\chi_{4477}(2538,\cdot)\)
\(\chi_{4477}(2659,\cdot)\)
\(\chi_{4477}(2756,\cdot)\)
...
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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 };
\((1333,3147)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{31}{36}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 4477 }(2538, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{180}\right)\) | \(e\left(\frac{71}{90}\right)\) | \(e\left(\frac{29}{90}\right)\) | \(e\left(\frac{1}{180}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{59}{90}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{9}\right)\) |
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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)
