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Properties

Label 1224.691
Modulus $1224$
Conductor $1224$
Order $48$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

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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(1224, base_ring=CyclotomicField(48)) M = H._module chi = DirichletCharacter(H, M([24,24,32,21]))
 
Copy content gp:[g,chi] = znchar(Mod(691, 1224))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1224.691");
 

Basic properties

Modulus: \(1224\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1224\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(48\)
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 1224.da

\(\chi_{1224}(139,\cdot)\) \(\chi_{1224}(211,\cdot)\) \(\chi_{1224}(283,\cdot)\) \(\chi_{1224}(403,\cdot)\) \(\chi_{1224}(499,\cdot)\) \(\chi_{1224}(547,\cdot)\) \(\chi_{1224}(571,\cdot)\) \(\chi_{1224}(619,\cdot)\) \(\chi_{1224}(643,\cdot)\) \(\chi_{1224}(691,\cdot)\) \(\chi_{1224}(787,\cdot)\) \(\chi_{1224}(907,\cdot)\) \(\chi_{1224}(979,\cdot)\) \(\chi_{1224}(1051,\cdot)\) \(\chi_{1224}(1195,\cdot)\) \(\chi_{1224}(1219,\cdot)\)

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: \(\Q(\zeta_{48})\)
Fixed field: Number field defined by a degree 48 polynomial

Values on generators

\((919,613,137,649)\) → \((-1,-1,e\left(\frac{2}{3}\right),e\left(\frac{7}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 1224 }(691, a) \) \(1\)\(1\)\(e\left(\frac{1}{48}\right)\)\(e\left(\frac{47}{48}\right)\)\(e\left(\frac{35}{48}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{19}{48}\right)\)\(e\left(\frac{1}{24}\right)\)\(e\left(\frac{41}{48}\right)\)\(e\left(\frac{37}{48}\right)\)\(1\)
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)
 
\( \chi_{ 1224 }(691,a) \;\) at \(\;a = \) e.g. 2