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Properties

Label 416.t
Modulus $416$
Conductor $13$
Order $4$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(416, base_ring=CyclotomicField(4)) M = H._module chi = DirichletCharacter(H, M([0,0,3])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(161,416)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(416\)
Conductor: \(13\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(4\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from 13.d
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Related number fields

Field of values: \(\mathbb{Q}(i)\)
Fixed field: 4.0.2197.1

Characters in Galois orbit

Character \(-1\) \(1\) \(3\) \(5\) \(7\) \(9\) \(11\) \(15\) \(17\) \(19\) \(21\) \(23\)
\(\chi_{416}(161,\cdot)\) \(-1\) \(1\) \(1\) \(-i\) \(i\) \(1\) \(i\) \(-i\) \(-1\) \(-i\) \(i\) \(-1\)
\(\chi_{416}(385,\cdot)\) \(-1\) \(1\) \(1\) \(i\) \(-i\) \(1\) \(-i\) \(i\) \(-1\) \(i\) \(-i\) \(-1\)