Properties

Label 3724.5
Modulus $3724$
Conductor $931$
Order $126$
Real no
Primitive no
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(126))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,87,112]))
 
pari: [g,chi] = znchar(Mod(5,3724))
 

Basic properties

Modulus: \(3724\)
Conductor: \(931\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(126\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{931}(5,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3724.ee

\(\chi_{3724}(5,\cdot)\) \(\chi_{3724}(101,\cdot)\) \(\chi_{3724}(213,\cdot)\) \(\chi_{3724}(397,\cdot)\) \(\chi_{3724}(453,\cdot)\) \(\chi_{3724}(465,\cdot)\) \(\chi_{3724}(537,\cdot)\) \(\chi_{3724}(633,\cdot)\) \(\chi_{3724}(745,\cdot)\) \(\chi_{3724}(929,\cdot)\) \(\chi_{3724}(985,\cdot)\) \(\chi_{3724}(997,\cdot)\) \(\chi_{3724}(1069,\cdot)\) \(\chi_{3724}(1165,\cdot)\) \(\chi_{3724}(1277,\cdot)\) \(\chi_{3724}(1461,\cdot)\) \(\chi_{3724}(1517,\cdot)\) \(\chi_{3724}(1529,\cdot)\) \(\chi_{3724}(1601,\cdot)\) \(\chi_{3724}(1809,\cdot)\) \(\chi_{3724}(1993,\cdot)\) \(\chi_{3724}(2049,\cdot)\) \(\chi_{3724}(2061,\cdot)\) \(\chi_{3724}(2133,\cdot)\) \(\chi_{3724}(2229,\cdot)\) \(\chi_{3724}(2341,\cdot)\) \(\chi_{3724}(2525,\cdot)\) \(\chi_{3724}(2581,\cdot)\) \(\chi_{3724}(2593,\cdot)\) \(\chi_{3724}(2761,\cdot)\) ...

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: $\Q(\zeta_{63})$
Fixed field: Number field defined by a degree 126 polynomial (not computed)

Values on generators

\((1863,3041,3137)\) → \((1,e\left(\frac{29}{42}\right),e\left(\frac{8}{9}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(23\)\(25\)\(27\)
\(-1\)\(1\)\(e\left(\frac{31}{126}\right)\)\(e\left(\frac{31}{126}\right)\)\(e\left(\frac{31}{63}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{29}{126}\right)\)\(e\left(\frac{31}{63}\right)\)\(e\left(\frac{19}{126}\right)\)\(e\left(\frac{1}{63}\right)\)\(e\left(\frac{31}{63}\right)\)\(e\left(\frac{31}{42}\right)\)
value at e.g. 2