Properties

Label 3724.2007
Modulus $3724$
Conductor $3724$
Order $42$
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(42))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([21,5,35]))
 
pari: [g,chi] = znchar(Mod(2007,3724))
 

Basic properties

Modulus: \(3724\)
Conductor: \(3724\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
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.dl

\(\chi_{3724}(255,\cdot)\) \(\chi_{3724}(787,\cdot)\) \(\chi_{3724}(943,\cdot)\) \(\chi_{3724}(1319,\cdot)\) \(\chi_{3724}(1475,\cdot)\) \(\chi_{3724}(1851,\cdot)\) \(\chi_{3724}(2007,\cdot)\) \(\chi_{3724}(2539,\cdot)\) \(\chi_{3724}(2915,\cdot)\) \(\chi_{3724}(3071,\cdot)\) \(\chi_{3724}(3447,\cdot)\) \(\chi_{3724}(3603,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 42 polynomial

Values on generators

\((1863,3041,3137)\) → \((-1,e\left(\frac{5}{42}\right),e\left(\frac{5}{6}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(23\)\(25\)\(27\)
\(-1\)\(1\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{13}{42}\right)\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{5}{14}\right)\)
value at e.g. 2