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Properties

Label 403.bf
Modulus $403$
Conductor $403$
Order $12$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(403, base_ring=CyclotomicField(12)) M = H._module chi = DirichletCharacter(H, M([7,10])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(37,403)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(403\)
Conductor: \(403\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(12\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: yes
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Related number fields

Field of values: \(\Q(\zeta_{12})\)
Fixed field: 12.12.1468905353759383637213883637.2

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(7\) \(8\) \(9\) \(10\) \(11\)
\(\chi_{403}(37,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{12}\right)\) \(-i\) \(-i\) \(-i\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(i\)
\(\chi_{403}(150,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{7}{12}\right)\) \(-i\) \(-i\) \(-i\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(i\)
\(\chi_{403}(223,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{12}\right)\) \(i\) \(i\) \(i\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(-i\)
\(\chi_{403}(305,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{12}\right)\) \(i\) \(i\) \(i\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(-i\)