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Properties

Label 1247.380
Modulus $1247$
Conductor $1247$
Order $84$
Real no
Primitive yes
Minimal yes
Parity odd

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(1247, base_ring=CyclotomicField(84)) M = H._module chi = DirichletCharacter(H, M([15,28]))
 
Copy content gp:[g,chi] = znchar(Mod(380, 1247))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1247.380");
 

Basic properties

Modulus: \(1247\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1247\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(84\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 1247.dt

\(\chi_{1247}(79,\cdot)\) \(\chi_{1247}(135,\cdot)\) \(\chi_{1247}(221,\cdot)\) \(\chi_{1247}(251,\cdot)\) \(\chi_{1247}(264,\cdot)\) \(\chi_{1247}(337,\cdot)\) \(\chi_{1247}(350,\cdot)\) \(\chi_{1247}(380,\cdot)\) \(\chi_{1247}(466,\cdot)\) \(\chi_{1247}(479,\cdot)\) \(\chi_{1247}(565,\cdot)\) \(\chi_{1247}(595,\cdot)\) \(\chi_{1247}(681,\cdot)\) \(\chi_{1247}(694,\cdot)\) \(\chi_{1247}(780,\cdot)\) \(\chi_{1247}(810,\cdot)\) \(\chi_{1247}(823,\cdot)\) \(\chi_{1247}(896,\cdot)\) \(\chi_{1247}(909,\cdot)\) \(\chi_{1247}(939,\cdot)\) \(\chi_{1247}(1025,\cdot)\) \(\chi_{1247}(1081,\cdot)\) \(\chi_{1247}(1197,\cdot)\) \(\chi_{1247}(1210,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((1162,175)\) → \((e\left(\frac{5}{28}\right),e\left(\frac{1}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1247 }(380, a) \) \(-1\)\(1\)\(e\left(\frac{5}{28}\right)\)\(e\left(\frac{19}{84}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{15}{28}\right)\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{37}{84}\right)\)\(e\left(\frac{13}{28}\right)\)
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)
 
\( \chi_{ 1247 }(380,a) \;\) at \(\;a = \) e.g. 2