Properties

Label 3724.531
Modulus $3724$
Conductor $3724$
Order $14$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(3724, base_ring=CyclotomicField(14))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([7,5,7]))
 
pari: [g,chi] = znchar(Mod(531,3724))
 

Basic properties

Modulus: \(3724\)
Conductor: \(3724\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(14\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3724.by

\(\chi_{3724}(531,\cdot)\) \(\chi_{3724}(1063,\cdot)\) \(\chi_{3724}(1595,\cdot)\) \(\chi_{3724}(2127,\cdot)\) \(\chi_{3724}(2659,\cdot)\) \(\chi_{3724}(3191,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{7})\)
Fixed field: 14.0.19640210869081831858492466681823232.1

Values on generators

\((1863,3041,3137)\) → \((-1,e\left(\frac{5}{14}\right),-1)\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(23\)\(25\)\(27\)
\(-1\)\(1\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{1}{14}\right)\)
value at e.g. 2