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Properties

Label 18496.3099
Modulus $18496$
Conductor $18496$
Order $272$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(18496, base_ring=CyclotomicField(272)) M = H._module chi = DirichletCharacter(H, M([136,153,37]))
 
Copy content gp:[g,chi] = znchar(Mod(3099, 18496))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("18496.3099");
 

Basic properties

Modulus: \(18496\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(18496\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(272\)
Copy content comment:Order
 
Copy content sage:chi.multiplicative_order()
 
Copy content gp:charorder(g,chi)
 
Copy content magma:Order(chi);
 
Real: no
Copy content comment:Whether the character is real
 
Copy content sage:chi.multiplicative_order() <= 2
 
Copy content gp:charorder(g,chi) <= 2
 
Copy content magma:Order(chi) le 2;
 
Primitive: yes
Copy content comment:If the character is primitive
 
Copy content sage:chi.is_primitive()
 
Copy content gp:#znconreyconductor(g,chi)==1
 
Copy content magma:IsPrimitive(chi);
 
Minimal: yes
Parity: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 18496.hb

\(\chi_{18496}(139,\cdot)\) \(\chi_{18496}(211,\cdot)\) \(\chi_{18496}(227,\cdot)\) \(\chi_{18496}(235,\cdot)\) \(\chi_{18496}(435,\cdot)\) \(\chi_{18496}(923,\cdot)\) \(\chi_{18496}(963,\cdot)\) \(\chi_{18496}(1083,\cdot)\) \(\chi_{18496}(1227,\cdot)\) \(\chi_{18496}(1299,\cdot)\) \(\chi_{18496}(1315,\cdot)\) \(\chi_{18496}(1323,\cdot)\) \(\chi_{18496}(1523,\cdot)\) \(\chi_{18496}(2011,\cdot)\) \(\chi_{18496}(2051,\cdot)\) \(\chi_{18496}(2171,\cdot)\) \(\chi_{18496}(2315,\cdot)\) \(\chi_{18496}(2403,\cdot)\) \(\chi_{18496}(2411,\cdot)\) \(\chi_{18496}(2611,\cdot)\) \(\chi_{18496}(3099,\cdot)\) \(\chi_{18496}(3259,\cdot)\) \(\chi_{18496}(3475,\cdot)\) \(\chi_{18496}(3491,\cdot)\) \(\chi_{18496}(3499,\cdot)\) \(\chi_{18496}(3699,\cdot)\) \(\chi_{18496}(4187,\cdot)\) \(\chi_{18496}(4227,\cdot)\) \(\chi_{18496}(4347,\cdot)\) \(\chi_{18496}(4491,\cdot)\) ...

Copy content comment:Galois orbit
 
Copy content sage:chi.galois_orbit()
 
Copy content gp:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 
Copy content magma:order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Related number fields

Field of values: $\Q(\zeta_{272})$
Fixed field: Number field defined by a degree 272 polynomial (not computed)

Values on generators

\((17919,1157,17921)\) → \((-1,e\left(\frac{9}{16}\right),e\left(\frac{37}{272}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(19\)\(21\)\(23\)
\( \chi_{ 18496 }(3099, a) \) \(1\)\(1\)\(e\left(\frac{11}{34}\right)\)\(e\left(\frac{97}{136}\right)\)\(e\left(\frac{57}{272}\right)\)\(e\left(\frac{11}{17}\right)\)\(e\left(\frac{15}{34}\right)\)\(e\left(\frac{27}{272}\right)\)\(e\left(\frac{5}{136}\right)\)\(e\left(\frac{93}{272}\right)\)\(e\left(\frac{145}{272}\right)\)\(e\left(\frac{193}{272}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x)
 
Copy content gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
 
Copy content magma:chi(x)
 
\( \chi_{ 18496 }(3099,a) \;\) at \(\;a = \) e.g. 2