Properties

Label 3724.167
Modulus $3724$
Conductor $3724$
Order $126$
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(126))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([63,117,77]))
 
pari: [g,chi] = znchar(Mod(167,3724))
 

Basic properties

Modulus: \(3724\)
Conductor: \(3724\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(126\)
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.ej

\(\chi_{3724}(167,\cdot)\) \(\chi_{3724}(223,\cdot)\) \(\chi_{3724}(279,\cdot)\) \(\chi_{3724}(307,\cdot)\) \(\chi_{3724}(363,\cdot)\) \(\chi_{3724}(447,\cdot)\) \(\chi_{3724}(699,\cdot)\) \(\chi_{3724}(755,\cdot)\) \(\chi_{3724}(811,\cdot)\) \(\chi_{3724}(839,\cdot)\) \(\chi_{3724}(895,\cdot)\) \(\chi_{3724}(1231,\cdot)\) \(\chi_{3724}(1287,\cdot)\) \(\chi_{3724}(1343,\cdot)\) \(\chi_{3724}(1427,\cdot)\) \(\chi_{3724}(1511,\cdot)\) \(\chi_{3724}(1819,\cdot)\) \(\chi_{3724}(1875,\cdot)\) \(\chi_{3724}(1903,\cdot)\) \(\chi_{3724}(2043,\cdot)\) \(\chi_{3724}(2295,\cdot)\) \(\chi_{3724}(2407,\cdot)\) \(\chi_{3724}(2435,\cdot)\) \(\chi_{3724}(2491,\cdot)\) \(\chi_{3724}(2575,\cdot)\) \(\chi_{3724}(2827,\cdot)\) \(\chi_{3724}(2883,\cdot)\) \(\chi_{3724}(2967,\cdot)\) \(\chi_{3724}(3023,\cdot)\) \(\chi_{3724}(3107,\cdot)\) ...

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

\((1863,3041,3137)\) → \((-1,e\left(\frac{13}{14}\right),e\left(\frac{11}{18}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(23\)\(25\)\(27\)
\(-1\)\(1\)\(e\left(\frac{47}{126}\right)\)\(e\left(\frac{89}{126}\right)\)\(e\left(\frac{47}{63}\right)\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{44}{63}\right)\)\(e\left(\frac{5}{63}\right)\)\(e\left(\frac{41}{126}\right)\)\(e\left(\frac{1}{126}\right)\)\(e\left(\frac{26}{63}\right)\)\(e\left(\frac{5}{42}\right)\)
value at e.g. 2